A landing theorem for entire functions with bounded post-singular sets

The Douady-Hubbard landing theorem for periodic external rays is one of the cornerstones of the study of polynomial dynamics. It states that, for a complex polynomial with bounded postcritical set, every periodic external ray lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic point is the landing point of at least one periodic external ray. We prove an analogue of this theorem for an entire function with bounded postsingular set. If the function has finite order of growth, then it is known that the escaping set contains certain curves called"periodic hairs"; we show that every periodic hair lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic periodic point is the landing point of at least one periodic hair. For a postsingularly bounded entire function of infinite order, such hairs may not exist. Therefore we introduce certain dynamically natural connected sets, called"filaments". We show that every periodic filament lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic periodic point is the landing point of at least one periodic filament. More generally, we prove that every point of a hyperbolic set is the landing point of a filament.


A note on terminology by the second author
In previous versions of this article, including that published in 2020 in Geometric and Functional Analysis, the objects we now call "filaments" were called "dreadlocks" -a term that I was using informally to refer to generalisations of "hairs" for some time before collaboration on the present paper began. I now regret this terminology. Dreadlocks are a hairstyle particularly associated with afro-textured hair. I was ignorant of the shameful history of discrimination against those whose hair does not conform to Eurocentric beauty standards; see "How the fight for natural black hair became a civil rights issue" (Keli Godd for The Guardian, April 2021). While I find the mathematical objects in question beautiful and natural, some may consider them exotic or pathological. This makes the term "dreadlocks" particularly inappropriate; it has been replaced with the agreement of my co-author. I encourage all mathematicians to be mindful of terminology that may be offensive or unwelcoming to colleagues from underrepresented groups.

Introduction
Let p : C → C be a polynomial. The filled-in Julia set K(p) consists of those points z ∈ C whose orbits remain bounded under repeated application of p. In their study of the dynamics of complex polynomials and the Mandelbrot set [DH85], Douady and Hubbard introduced the notion of external rays, which can be characterised as the gradient lines of the Green's function on the basin of attraction of infinity, C \ K(p). Periodic (and pre-periodic) rays are of particular importance, due to the following result. is bounded. (Equivalently, assume that K(p) is connected.) Then every periodic ray of p lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic periodic point of p is the landing point of at least one and at most finitely many periodic external rays.
The first half of this theorem, concerning the landing of periodic rays, can be found in [DH85, Exposé VIII.II, Proposition 2]. The second half, which is more difficult, is due to Douady; the first published proofs are in [EL89,Hub93]. Ever since, the Douady-Hubbard theorem has been a cornerstone of the study of polynomial dynamics. In particular, it forms the basis of the "puzzle techniques" that were pioneered by Yoccoz, Branner and Hubbard, and continue to lead to fundamental new results; see [Hub93,RY08,ALS11].
In the study of rational functions and transcendental entire functions, there is no immediate analogue of the basin of infinity, and this is one of the reasons that the study of these classes has presented greater challenges than that of polynomials. Nonetheless, in both settings analogues of the above-mentioned puzzle techniques have been employed to certain classes of functions with considerable success. We refer to [Roe08,Ben15] for two examples.
Our goal is to extend the Douady-Hubbard landing theorem to the case of a transcendental entire function f . In this setting, the role that critical values play in polynomial dynamics is taken by the larger set S(f ) of singular values of f . These are those points not having a neighbourhood in which all branches of f −1 are defined and holomorphic. Analogously to (1.1), the postsingular set of f is defined as For transcendental maps, ∞ is an essential singularity, rather than a super-attracting fixed point. Hence the definition of external rays for polynomials as gradient lines of a Green's function has no natural analogue. Nonetheless, it has long been known that the escaping set I(f ) . . = {z ∈ C : f n (z) → ∞} often contains curves to infinity; indeed, in some cases this was already noticed by Fatou [Fat26]. It was the work of Devaney and his collaborators (see e.g. [DK84,DT86]) that really began the study of these hairs or dynamic rays in the 1980s, particularly for functions in the exponential family, (1.3) f a : z → e z + a.
Devaney, Goldberg and Hubbard were probably the first to suggest that such hairs could serve as analogues of external rays of polynomials; compare [DGH86, BDH + 99]. Subsequently, Schleicher and Zimmer [SZ03a] and Schleicher and Rottenfußer [RS08b] proved that, for the families of exponential maps (1.3) and of cosine maps z → ae z +be −z , respectively, the entire escaping set I(f ) consists of hairs.
On the other hand, in [RRRS11] it is shown that there is a transcendental entire function f for which I(f ) contains no arcs. Hence there are no curves in I(f ), of any kind, landing at any of the repelling periodic points of f . (Recall that repelling periodic points are dense in the Julia set of any transcendental entire function.) Furthermore, the postsingular set P(f ) of this function is bounded. Indeed, S(f ) is a compact subset of the immediate basin of attraction of a single attracting fixed point.
Filaments. In view of the preceding example, we develop a novel approach to the landing problem that removes the focus on hairs altogether, by connecting repelling periodic points to infinity using more general sets of escaping points.
More precisely, we introduce a notion of filaments for postsingularly bounded entire functions. These are certain unbounded connected sets of escaping points generalising the concept of hairs. (See Section 4 for formal definitions.) The set of filaments has a natural combinatorial structure, and in tame cases, all filaments are in fact hairs. In general, however, filaments can be topologically much more complicated. Indeed, it follows from [Rem16] that the closure of a filament may be a hereditrarily indecomposable continuum. 1 With this terminology, we are able to prove the following generalisation of the Douady-Hubbard landing theorem for postsingularly bounded entire functions: Every periodic filament lands, and every repelling or parabolic periodic point is the landing point of at least one and at most finitely many periodic filaments (Theorem 8.1). In particular, without requiring the definitions of Section 4, we can state the following result.
Theorem 1.1 (Landing at periodic points). Let f be a transcendental entire function such that P(f ) is bounded, and let ζ be a repelling or parabolic periodic point. Then there is a connected and unbounded set A ⊂ I(f ) and a period p with the following properties.
(a) A = A ∪ {ζ} and A does not separate the plane; (b) f p (A) = A and f j (A) ∩ A = ∅ for 1 ≤ j < p; (c) for every ε > 0, f n tends to ∞ uniformly on {z ∈ A : |z − ζ| ≥ ε}. Ifζ = ζ is a different repelling or parabolic periodic point andÃ is a set as above forζ, then A ∩Ã = ∅.
We emphasise that using filaments, rather than restricting to cases where hairs exist (see Theorem 1.4 below), is crucial if one wishes to obtain results for general classes of functions. Indeed, Pfrang [Pfr19] uses our results to construct (homotopy) Hubbard trees for all postsingularly finite entire functions. This is a natural result whose hypothesis and conclusion make no mention of hairs. Its proof in this form is made possible by the use of filaments; compare the discussion at the end of the final section of [PRS18]. Similarly, work of Fagella and the first author [BF15,BF17,BF20], was formulated only for functions with hairs, but contains a number of results whose conclusion makes sense without this assumption. For example, the conclusion of the main theorem of [BF17] states that every non-repelling cycle has a singular orbit that is associated to it in a certain explicit manner. These results should now extend to all postsingularly bounded entire functions, by replacing the role of hairs in the proofs by our "filaments". In addition, the key technique of fundamental tails that we use to control filaments (see Section 3) has already found further applications, for instance in the study of inner functions arising in transcendental dynamics [EFJS19], and in a new version of the Fatou-Shishikura inequality [BF20].
Moreover, our results offer the possibility of developing puzzle-type arguments for all postsingularly bounded entire functions, and of using the powerful techniques of symbolic dynamics to study the behaviour of non-escaping points. As mentioned above, this is the reason why the structure of polynomial Julia sets is so well understood. Theorem 1.1 opens up large classes of entire transcendental functions to the same type of analysis.
Existence and landing of periodic hairs. In many interesting cases, periodic filaments are in fact periodic hairs. That is, the connected set A in Theorem 1.1 is an arc connecting ζ to ∞. In particular, this holds for functions satisyfing the following property, which states that the escaping set consists entirely of hairs.
Definition 1.2 (Criniferous functions). We say that an entire function f is criniferous 2 if the following holds for every z ∈ I(f ): For all sufficiently large n there is an arc γ n connecting f n (z) to ∞, in such a way that f maps γ n injectively onto γ n+1 , and such that min z∈γn |z| → ∞ as n → ∞.
The counterexample from [RRRS11] mentioned above shows that entire functions, even those with bounded postsingular sets, need not be criniferous. However, the same article also establishes criniferousness for a large and natural class of functions, as follows. The Eremenko-Lyubich class B consists of those transcendental entire functions for which S(f ) is bounded, and hence compact. (If P(f ) is bounded, then f ∈ B by definition.) It is proved in [RRRS11] that f is criniferous whenever f ∈ B and f has finite order of growth, i.e., log log|f (z)| = O(log|z|). Furthermore, any finite composition of functions with these properties is also criniferous.
To discuss periodic hairs, let us use the following definition from [Rem08].
Definition 1.3 (Periodic hairs). An invariant hair of a transcendental entire function f is a continuous and injective curve γ : R → I(f ) such that f (γ(t)) = γ(t + 1) for all t and lim t→+∞ |γ(t)| = ∞. A periodic hair is a curve that is an invariant hair for some iterate f n of f .
Such a hair lands if the limit z 0 = lim t→−∞ γ(t) exists; this limit is called the landing point (sometimes also endpoint) of the hair γ.
With this terminology, Theorem 1.1 takes the following form for criniferous functions.
Theorem 1.4 (Landing theorem for periodic hairs). Let f be a transcendental entire function such that the postsingular set P(f ) is bounded. Then every periodic hair of f lands at a repelling or parabolic periodic point. If, in addition, f is criniferous, then conversely every repelling or parabolic periodic point of p is the landing point of at least one and at most finitely many periodic hairs.
The first part of the theorem, concerning landing behaviour of periodic rays, is not new. It was proved for exponential maps in [SZ03b], and later in full generality by the second author [Rem08,Corollary B.4]; see also [Den14]. The proof uses similar ideas as in the polynomial case, namely expansion properties for the hyperbolic metric, although there are also some additional ingredients.
On the other hand, the usual proofs for accessibility of repelling and parabolic periodic points in the polynomial case [EL89,Hub93,Prz94] strongly rely on the presence of the open basin of attraction of infinity, and thus break down completely in the transcendental setting. Nonetheless, there has been some previous work in this direction. Under the additional dynamical assumption that f is geometrically finite, the theorem was proved by Mihaljević-Brandt [Mih10]. Furthermore, the first author and Lyubich [BL14] proved Theorem 1.4 when f belongs to the exponential family (1.3).
For exponential maps, boundedness of the postsingular set is a strong dynamical condition (though weaker than geometrical finiteness), as it implies non-recurrence of the singular value a. However, the non-recurrence property is not used in any essential way in [BL14], and the ideas used there form one of the ingredients in our proofs of Theorems 1.1 and 1.4.
During the preparation of this manuscript, Dierk Schleicher informed us that he has an alternative approach to Theorem 1.4, using ideas from [SZ03b].
Hyperbolic sets. As in [BL14], our techniques apply not only to repelling (and parabolic) periodic points, but also to hyperbolic sets; see [Prz94] for the corresponding result for polynomials. Recall that a compact, forward-invariant set K ⊂ C is called hyperbolic if for some k ∈ N and η > 1 we have |(f k ) (z)| > η for all z ∈ K.
If P(f ) is bounded and K is such a hyperbolic set, then we prove that every point of K is "accessible" from the escaping set, via a filament (see Theorem 8.2). Again, we can state the following result without requiring the terminology of filaments.
Theorem 1.5 (Landing at hyperbolic sets). Let f be a transcendental entire function such that P(f ) is bounded, and let K be a hyperbolic set of f . Then there is a collection A of pairwise disjoint, connected and unbounded sets A ⊂ I(f ) with the following properties.
(a) For every A ∈ A, there is z 0 (A) ∈ K such that A = A ∪ {z 0 (A)}, and A does not separate the plane; (b) the function A → K; A → z 0 (A) is surjective; (c) f (A) ∈ A for all A ∈ A; (d) for every ε > 0, f n tends to ∞ uniformly on {z ∈ A : dist(z, K) ≥ ε}; (e) if z 0 (A) is periodic of period p, then f kp (A) = A for some k ≥ 1. If f is criniferous, then every A ∈ A is an arc connecting z 0 (A) ∈ K to ∞.
This generalisation is of particular relevance in the case where P(f ) itself is a hyperbolic set, which is often the case for non-recurrent entire functions (see [RvS11]). Hence, in this case, each singular value can itself be connected to infinity by a filament, which in turn allows one to study the Julia set via symbolic dynamics rather closely. For example, in [Ben15], the existence of a ray landing at the omitted value is exploited to prove strong rigidity properties of non-recurrent parameters in the exponential family, extending previous work [Ben11] in the postsingularly finite case.
To conclude the introduction, we remark on the case where the postsingular set P(f ) is unbounded. If f is a polynomial, then every unbounded orbit escapes to infinity. For polynomials with escaping singular orbits, the Douady-Hubbard landing theorem no longer holds. Indeed, it is possible that a repelling periodic point is the landing point of uncountably many external rays, none of which are periodic. Compare [LP96].
For transcendental entire functions, it is possible for singular orbits to be unbounded without converging to infinity. It is conceivable that, for f ∈ B with all singular orbits nonescaping, a version of the landing theorem holds. However, even for exponential maps this is not known (see [Rem06a] for a partial result), and it appears that significant further new ideas would be required to approach it. See Section 14 for further discussion.
Structure of the paper. Section 2 gives an overview of expansivity properties for functions in class B without the assumption of bounded postsingular set. It also defines the concept of external addresses, and gives sufficient conditions on such addresses to be realised by certain unbounded connected sets of points. Several of the ideas used in this section are already implicitly or explicitly contained in the literature, e.g. in [EL92,Rem07a,Rem08,Rem09], but are combined here in a novel, systematic and unified manner.
From Section 3 onward, we restrict to functions with bounded postsingular sets, beginning by discussing hyperbolic expansion estimates for such maps, and introducing the important combinatorial notion of fundamental tails. With these preparations, Section 4 introduces filaments for a function with bounded postsingular set, studies their main topological and combinatorial properties, and also shows that the escaping set consists of filaments. The ideas in this section have their roots in [Rem07a]. In particular, we recover the main result of that paper; see Corollary 4.11. In Section 5, we discuss the relation between filaments and hairs.
Section 6 introduces accumulation sets of filaments at bounded addresses, and gives different characterisations of when a filament lands. This section contains a crucial innovation, which is central to the proofs of our main theorems: Rather than having to contend with the potentially complicated topological structure of filaments, we can instead study their landing properties by considering a certain chain of open simply connected sets. In Section 7, we establish that such a landing filament cannot separate the plane. We are then ready to state our main theorems concerning filaments in Section 8, and to derive Theorems 1.1 and 1.4 from these.
The three following sections are dedicated to proving the main theorems of this paper. Section 9 establishes the landing of periodic filaments, in Section 10 we show accessibility of hyperbolic sets and repelling periodic orbits, and finally Section 11 is dedicated to the proof of accessibility of parabolic points.
We remark that one can take an alternative, less natural but more direct, approach to establishing our theorems, bypassing most of the material in Sections 2 and 4-6. Readers interested in such a short-cut are referred to Remark 8.4.
To round off the paper, Section 12 discusses bounds on the number of rays landing together at a given point in a hyperbolic set. We also include two appendices. The first, Section 13, gives some details concerning the cyclic order at infinity of unbounded connected sets, which are used in some of our arguments. The second, Section 14, discusses open questions about landing theorems for entire functions with unbounded postsingular sets.
Notation and preliminaries. We write C for the complex plane andĈ for the Riemann sphere. We denote the closure in C of a set A ⊂ C by A, and occasionally cl(A). The closure of A inĈ is denoted byÂ.
The Euclidean disk of radius R around a point z is denoted by D R (z); the unit disk is D . . = D 1 (0). If D is any Euclidean disc, we also write r(D) for the radius of D.
We denote Euclidean distance and diameter by dist and diam, respectively. If U ⊂ C is an open set omitting more than two points, then we denote hyperbolic distance on U by dist U , and similarly diam U for hyperbolic diameter. We also denote the density of the hyperbolic metric of U at a point z by ρ U (z). That is, the length element of the hyperbolic metric is given by ρ U (z)|dz|.
Acknowledgements. We are extremely grateful to Dave Sixsmith and to David Pfrang for their many and extraordinarily helpful suggestions that considerably improved the presentation of the paper. We also thank Daniel Meyer for interesting comments, particularly a suggestion on the presentation of cyclic order in Section 13.

Unbounded sets of escaping points
In this section, we briefly review basic properties of the dynamics of a function f ∈ B, and review the definition of external addresses for such maps. Then we state a theorem (Theorem 2.5) about the existence of unbounded connected sets for these addresses, and devote the rest of the section to the proof thereof. These sets will provide the basis of the "filaments" that are introduced (for postsingularly bounded entire functions) in Section 4.
Throughout this section, fix a function f ∈ B. Recall that this implies that S(f ) is bounded. For now, we do not assume that P(f ) is also bounded. Let us begin by reviewing the method of partitioning the locus where f is large into (topological) half-strips known as fundamental domains. (Compare e.g. [Rem08, Section 2] or [Rot05, Section 2].) For this construction, we fix a Euclidean disk D around the origin containing S(f ). The connected components of f −1 (C \ D) are called the tracts of f . If T is a tract, then f : T → C \ D is a universal covering map; in particular, T is unbounded and simply connected. In fact (applying the same argument to a slightly smaller disc than D), T is a Jordan domain inĈ whose boundary passes through infinity, and f is a universal covering f : T → C \ D on the closure of T (in C). Figure 1. The dynamical plane for a function with two tracts T 1 and T 2 . One of the fundamental domains F i obtained by taking preimages of δ is shown inside T 2 .
We may assume in the following that D ∩ f (D) = ∅, e.g. by ensuring that f (0) ∈ D. Then it is easy to see that there is an arc δ connecting a point of ∂D to infinity in the complement of the closure of the tracts. We define The connected components of f −1 (W 0 ) are called the fundamental domains of f ; see Figure 1. We remark that there are only finitely many fundamental domains that intersect a given compact set, due to the following simple fact.
Lemma 2.1 (Preimage components intersecting a compact set). Let f : X → Y be a holomorphic map between Riemann surfaces X and Y . Furthermore, let U ⊂ Y be a domain whose boundary (in Y ) is locally connected; i.e. every point of the boundary of U in Y has arbitrarily small connected relative neighbourhoods in Y .
Then for any compact set K ⊂ X, only finitely many connected components of f −1 (U ) intersect K.
Remark. The condition that ∂U is locally connected is necessary: Let X = Y = C, f = exp, and let U be a simply connected domain in the punctured unit disc that spirals in towards the unit circle. (I.e., any branch of the argument on U tends to infinity as |z| → 1 in U .) Then infinitely many components of f −1 (U ) intersect the closed unit disc.
Proof. We begin by reformulating the hypothesis that ∂U is locally connected, as follows.
Claim. Let ζ ∈ U , and let ∆ be a neighbourhood of ζ in Y . Then there is a finite collection of connected open sets W 1 , . . . , W n ⊂ V ∩ U such that {ζ} ∪ W 1 ∪ · · · ∪ W n is a neighbourhood of ζ in {ζ} ∪ U .
Proof. Shrinking ∆ if necessary, we may assume that ∆ is a closed topological disc, and that U ⊂ ∆. Consider the compact set Q . . = ∂∆ ∪ (∆ \ U ). Since U is connected but not contained in ∆, the boundary of each connected component of U ∩ ∆ = ∆ \ Q intersects ∂∆. Recall that ∂U is locally connected; it follows readily that Q is also. Hence U ∩ ∆ has at most finitely many connected components of diameter greater than, say, δ . . = dist(ζ, ∂∆)/2; see [Why42,Theorem 4.4 in Chapter VI]. (Here dist refers to distance with respect to some metric on the topological disc ∆.) Therefore only finitely many connected components of W 1 , . . . , W n of U ∩∆ intersect the disc of radius δ around ζ, as claimed.
Let z ∈ f −1 (U ), and ζ . . = f (z). Then z has a neighbourhood V 1 that is topologically mapped as by z → z d , where d is the local degree of f at z. Take W 1 , . . . , W n as in the claim, for ∆ = f (V 1 ). If V z ⊂ V 1 is a sufficiently small disc around z, then any point in f −1 (U ) ∩ V z maps into some W j . As f −1 (W j ) has d connected components in V 1 , it follows that V z intersects at most dn connected components of f −1 (U ).
So the compact set K ∩ f −1 (U ) has an open cover by sets V z , each of which intersects only finitely many connected components of f −1 (U ). The claim follows by taking a finite subcover.
It follows that there are only finitely many fundamental domains F whose closure intersects the disc D. When this does not occur for any F , the function f is dynamically particularly simple; more precisely, it is of disjoint type (hyperbolic with connected Fatou set). For a detailed study of the topological dynamics of such functions, see [Rem16]. (Compare also the discussion of disjoint-type addresses in Remark 4.15.) In the following, given a fundamental domain F we denote by ∞ F the unbounded connected component of F \ D.
Expansion properties and relative cylindrical distance. It is known that functions in B are strongly expanding near infinity. More precisely, if f ∈ B, then the cylindrical derivative of f is large whenever f (z) is large [EL92, Lemma 1]. That is, In view of (2.2), we may assume that the radius r(D) is chosen sufficiently large to ensure that In particular, f (0) ∈ D. These assumptions will remain in place for the remainder of the paper.
A number of results in the literature are phrased not for the function f directly, but in terms of a logarithmic transform L of f . (See e.g. [EL92] or [Rem07a].) Such a transform can be obtained using the change of variable z = ρ · exp(ζ), where ρ = r(D) is the radius of D. I.e., there is a 2πi-periodic function L defined by ρ · exp(L(ζ)) = f (ρ · exp(ζ)), defined whenever the right-hand side (i.e., f (z)) belongs to C \ D. Dynamical properties of L easily translate to properties of f on the set of points whose orbits remain outside  . In (a), the cylindrical distance between z and w is less than π. However, any curve connecting z and w in W 0 takes more than two full turns around D, and hence the relative distance in the sense of Definition 2.2 is greater than 4π. The second image, in (b) illustrates Lemma 2.7 in this case. Here Γ is a union of three cross-cuts of W 0 satisfying the conclusion of the lemma; the domain W Γ 0 is shown shaded in grey. (Nb. the set Γ constructed in the proof of Lemma 2.7 contains two additional crosscuts; omitting these does not change the domain W Γ 0 , and therefore the conclusions of the lemma.) D forever. Our assumptions on D imply that the function L is normalised in the sense of [Rem07a,RRRS11].
We occasionally cite results from other articles that are phrased in this language, but never use the logarithmic transform L directly in this article. Instead, we use the following terminology, which is inspired by this change of coordinates. See Figure 2 (a).
Definition 2.2 (Relative cylindrical distance). For z, w ∈ W 0 , we define the relative cylindrical distance dist W 0 cyl (z, w) to be the shortest cylindrical length of a curve γ from z to w that is homotopic, in C \ D, to a curve in W 0 .
Equivalently, if U is a connected component of exp −1 (W 0 ) and ζ, ω are the logarithms of z and w that belong to U , then We similarly define the distance between two subsets of W 0 , and the diameter diam W 0 cyl with respect to this metric.
Since f is expanding with respect to the cylindrical metric on W 0 by (2.3), we have cyl (z, w) whenever z and w both belong to ∞ F for some fundamental domain F . Let us also note the following fact for future reference.
n=0 is a sequence of points in W 0 , then |z n | → ∞ if and only if dist W 0 cyl (z n , ζ 0 ) → ∞. External addresses and symbolic dynamics. The reason for introducing fundamental domains is that they can be used to assign symbolic dynamics to points whose orbit stays sufficiently large, and in particular to escaping points.
Definition 2.4 (External addresses). Let f ∈ B, and let fundamental domains be defined as above. An (infinite) external address is a sequence s = F 0 F 1 F 2 . . . of fundamental domains of f . The address s is bounded if the set of fundamental domains occurring in s is finite; it is periodic if there is k such that F n+k = F n for all n ≥ 0.
Let s = F 0 F 1 F 2 . . . be an external address, and recall that ∞ F n denotes the unbounded connected component of F n \ D. Then we define Remark. For the purpose of this paper, we shall often use "address" synonymously with "external address".
The main goal of this section is to prove the following.
Theorem 2.5 (Realisation of addresses). Let s be an external address.
(a) Suppose that J 0 s contains some point z 0 . Then J 0 s also contains a closed unbounded connected set X on which the iterates of f tend to infinity uniformly. Moreover, dist W 0 cyl (z 0 , X) ≤ 4π. (b) If X 1 and X 2 are unbounded, closed, connected subsets of J 0 s with X 1 ⊂ X 2 , then X 2 ⊂ X 1 and f n | X 2 → ∞ uniformly. (c) If s is bounded, then J 0 s = ∅. Furthermore, there exists R > r(D), depending on the finite collection of fundamental domains occurring in s but not otherwise on s, such that the set X in (a) can be chosen to contain a point of modulus R.
(d) Conversely, if F is a finite collection of fundamental domains, then there is R > 0 such that the iterates of f tend to infinity uniformly on the closed set  [Rem09], it can then be deduced for general functions in the class B that J 0 s = ∅ for bounded s. Since the papers in question all use slightly different notation, we shall give a new proof of Theorem 2.5 that is self-contained and unified. We begin with a simple property of the set J 0 s , which is similar to [EL92, Theorem 1].
Lemma 2.6 (One-dimensionality of J 0 s ). Let s = F 0 F 1 . . . be an external address. Then J 0 s is a subset of J(f ), has empty interior and does not separate the plane.
Proof. For each n ≥ 0, U n .
It follows inductively that f n (V 0 ) ⊂ ∞ F n for all n ≥ 0. On the other hand, fix z 0 ∈ V 0 , and set z n . . = f n (z 0 ). It follows from the above that f n : V 0 → f n (V 0 ) is univalent for all n. By (2.4) and the definition of the cylindrical derivative, we have |(f n ) (z 0 )|/|f n (z 0 )| → ∞ as n → ∞. By Koebe's 1/4-theorem it follows that 0 ∈ f n (V 0 ) for sufficiently large n. This is a contradiction and proves that J 0 s has empty interior and does not separate the plane.
Let z ∈ J 0 s . Suppose first that dist(f n (z), D) → ∞ as n ≥ 0. If z belonged to the Fatou set of f , then it would follow from equicontinuity that there are n 0 and a neighbourhood V of z such that f n (V ) ∩ D = ∅ for all n ≥ n 0 . But then f n 0 (V ) ⊂ J 0 σ n 0 (s) , and this contradicts the result we have just proved. So z ∈ J(f ).
Otherwise, there is a sequence n k such that f n k (z) → ∞. Then the spherical derivative of f n k at z is comparable to the corresponding cylindrical derivative. By (2.4), the latter tends to infinity as k → ∞. Thus the family of iterates of f is not normal at z by Marty's theorem, and again z ∈ J(f ).
A separation lemma. We now formulate a key technical lemma -closely related to Lemmas 3.1 and 3.3 of [Rem07a] -that will be crucial for our unified proof of Theorem 2.5. Before making the formal statement, which is slightly technical, let us explain the idea. Let z 0 ∈ W 0 , and suppose that z 0 can be connected to infinity within the set of points of modulus greater than R. Then our lemma states (in particular) that no point ζ of modulus at most R can be connected to infinity without passing near z 0 , in the sense of relative cylindrical distance.
It may appear at first as though this is obvious, since the round circle centred at 0 and of modulus |z 0 | has cylindrical length 2π, and must intersect any curve connecting ζ to infinity. However, the diameter of the intersection of this circle with W 0 may be arbitrarily large when measured with respect to dist W 0 cyl ; see Figure 2. Lemma 2.7 (Cross-cuts of W 0 ). Let z 0 ∈ W 0 . Then there exists a union Γ z 0 of cross-cuts of W 0 such that dist W 0 cyl (z 0 , ζ) ≤ 2π for all ζ ∈ Γ and such that the unbounded connected component W 0 Γ of W 0 \ Γ has the following property. If R > 0 is such that z 0 belongs to the unbounded connected component of W 0 \ D R (0), then W Γ 0 is also disjoint from D R (0). Here Γ can be chosen to consist only of arcs of the circle of radius |z 0 | centred at the origin.
Moreover, suppose that A ⊂ W 0 is any unbounded connected set with z 0 ∈ A. Then, for all z ∈ W Γ 0 , dist W 0 cyl (z, A) ≤ 2π. Remark. The curve δ in the definition of fundamental domains can be chosen to be piecewise analytic, in which case the number of cross-cuts in Γ is necessarily finite. However, we do not require this.
Proof. Let U be a connected component of exp −1 (W 0 ); then exp : U → W 0 is a conformal isomorphism. For ζ ∈ U , let I ζ denote the vertical segment ζ + i · [−2π, 2π]. The fact that U is disjoint from its 2πiZ-translates implies the following separation property: if ζ 0 , ζ 1 ∈ U with ζ 0 / ∈ I ζ 1 , then either I ζ 1 separates ζ 0 from infinity in U , or vice versa.
Indeed, suppose otherwise. Then for j = 0, 1, there is a curve γ j ⊂ U connecting ζ j to infinity, and not intersecting I ζ 1−j . Set A . . = γ 0 ∪ γ 1 ∪ {∞}, and let a ∈ A be a point of minimal real part; say a ∈ γ j . By [Rem16, Corollary 5.4], I ζ separates a from ∞ for every ζ ∈ A \ I ζ . This is a contradiction to the fact that ζ 1−j ∈ A, but γ j does not intersect I ζ 1−j . Now let ζ 0 be the unique point of exp −1 (z 0 ) ∩ U and set I . . = I ζ 0 . Observe that the endpoints of I are elements of exp −1 (z 0 ) different from ζ 0 , and hence do not belong to U . We set X . . = I ∩ U ; then X is a collection of cross-cuts of U . Set Γ . . = exp(X); we will prove the claims of the lemma by considering the unbounded connected component V 1 of U \ I. In other words, V 1 consists of all points of U that are not separated from ∞ by I ζ .
By the separation property, for all ζ ∈ V 1 , I ζ separates ζ 0 from infinity in U . So if A ⊂ U is an unbounded connected set with ζ 0 ∈ A, then I ζ ∩ A = ∅, and in particular dist(ζ, A) ≤ 2π. Moreover, by hypothesis ζ 0 can be connected to infinity by a curve in U that stays at real parts greater than log R. So Re ζ > log R for all ζ ∈ V 1 .
Since exp(V 1 ) = W Γ 0 , this completes the proof of the lemma.
Existence of unbounded sets of escaping points. With these preparations, we are now able to prove the existence of unbounded connected subsets of J 0 s under very general hypotheses.
Theorem 2.8 (Unbounded subsets of J 0 s ). Let s = F 0 F 1 F 2 . . . be an external address. Suppose that (z n ) ∞ n=0 is a sequence of points such that each z n ∈ ∞ F n for all n ≥ 0. Suppose furthermore that there is Then there is a closed, unbounded and connected set X ⊂ J 0 s such that dist W 0 cyl (z 0 , X) ≤ 2 max(2π, C).
Proof. For n ≥ 0, let Γ n be the union of cross-cuts from Lemma 2.7, applied to z n . Let A k n be a sequence of unbounded closed connected sets, defined for n, k ≥ 0 as follows. Let A 0 n be the closure of the unbounded connected component of F n \ Γ n . For k ≥ 0, let A k+1 n be the closure of the unbounded connected component of (f | Fn ) −1 (A k n+1 ) \ Γ n . Observe that A k+1 n ⊂ A k n for all n and k. Claim. dist W 0 cyl (A k n , z n ) ≤ 2 max(2π, C) for all n, k ≥ 0.
Proof. Since A 0 n intersects Γ n , the claim is true for k = 0. Suppose that k ≥ 0 is such that the claim holds for all n ≥ 0. Let n ≥ 0, and set B . . = (f | Fn ) −1 (A k n+1 ). If B ∩Γ n = ∅, then A k+1 n intersects Γ n and the claim is immediate from the properties of Γ n . Otherwise, By the inductive hypothesis, and since Γ n separates ∂D from B, we can connect B to Γ n by a curve γ with diam W 0 cyl (γ) ≤ max(2π, C). Since diam W 0 cyl (Γ n ) ≤ 2π, the claim follows.
Now set X 1 . . = k≥0 A k 0 . Then X 1 ∪ {∞} is compact and connected as a countable intersection of compact connected sets. Moreover, X 1 contains a point ζ with dist W 0 cyl (ζ, z 0 ) ≤ 2 max(2π, C). If X is the connected component of X 1 containing ζ, then X is unbounded by the boundary bumping theorem [Nad92, Theorem 5.6], and the proof is complete.
In particular, we obtain the following partial results towards Theorem 2.5.
Corollary 2.9 (Realised addresses have unbounded sets). Suppose that s is an external address, and that there is a point z 0 ∈ J 0 s . Then J 0 s contains an unbounded closed connected set X, and dist W 0 cyl (z 0 , X) ≤ 4π.
Corollary 2.10 (Bounded addresses are realised). Suppose that F is a finite collection of fundamental domains. Then there is R > 0 with the following property. If s is an external address whose entries are all in F, then J 0 s contains an unbounded connected set X which contains a point of modulus at most R.
Proof. Pick a base-point ζ 0 ∈ W 0 . For F ∈ F, let ζ F be the preimage of ζ 0 in F . Then there is a constant C such that dist W 0 cyl (ζ 0 , ζ F ) ≤ C for all F ∈ F (when defined). If s is an external address in F N 0 , we can set z n . . = ζ 0 for all n ≥ 0, and apply Theorem 2.8. We obtain an unbounded connected set X with dist W 0 cyl (ζ 0 , X) ≤ 2 max(2π, C). If R is sufficiently large (depending only on C), then X contains a point of modulus at most R.
Uniform escape to infinity. To complete the proof of Theorem 2.5, we consider the question of uniform escape to infinity on an unbounded connected subset of J 0 s . Recall that by definition of J 0 s a point z belongs to J 0 s if and only if z n ∈ ∞ F n for all n, where s = F 0 F 1 . . .. Lemma 2.11 (Uniform escape on unbounded connected sets). Let s = F 0 F 1 F 2 . . . be an external address, and suppose that X ⊂ J 0 s is unbounded and connected. Furthermore, assume that there is a sequence (z n ) ∞ n=0 (not necessarily an orbit of f ) such that z n ∈ ∞ F n , and such that dist W 0 cyl (z n , f n (X)) → ∞. Then f n | X → ∞ uniformly.
Proof. Let ζ 0 ∈ W 0 be any base point. We may assume that dist W 0 cyl (z n , ζ 0 ) → ∞. Indeed, set η n . . = dist W 0 cyl (z n , f n (X)) and let (ξ n ) ∞ n=0 be any sequence in Clearly dist(z n , ζ 0 ) → ∞ and dist W 0 cyl (z n , f n (X)) ≥ η n /3 → ∞, as desired. Let Γ n be the union of cross-cuts associated to z n by Lemma 2.7. Then, for sufficiently large n, f n (X) is disjoint from Γ n , and hence belongs to W Γn 0 . Since |z n | → ∞ by Observation 2.3, it follows that f n (X) → ∞ uniformly, as required.
Proof of Theorem 2.5. We first prove (b), so let X 1 , X 2 ⊂ J 0 s be closed, unbounded and connected with X 1 ⊂ X 2 .
Let n 0 be sufficiently large that dist W 0 cyl (z n , f n (X 2 )) > 2π for n ≥ n 0 . Let Γ n be the union of crosscuts associated to z n by Lemma 2.7. Then f n (X 2 ) ⊂ W Γn 0 for n ≥ n 0 . Since f n (X 1 ) connects z n to ∞, we have dist(f n (X 1 ), f n (ζ)) ≤ 2π for all ζ ∈ X 2 and all n ≥ 0. Again by (2.4), we have dist(X 1 , ζ) = 0, and hence ζ ∈ X 1 , as required. Now let us prove (a), so suppose that z 0 ∈ J 0 s . By Corollary 2.9, there is an unbounded closed connected set X ⊂ J 0 s with dist W 0 cyl (X, z 0 ) ≤ 4π. We may assume that z 0 / ∈ X. Indeed, otherwise we let ε be sufficiently small and replace X by an unbounded connected component of X \ D ε (z 0 ) that intersects ∂D ε (z 0 ). Now set z n . . = f n (z 0 ). By (2.4), we have dist W 0 cyl (f n (X), z n ) → ∞, and hence it follows from Lemma 2.11 that f n | X → ∞ uniformly. This completes the proof of (a) of Theorem 2.5.
Part (c) follows directly from Corollary 2.10. To prove (d), observe first that equality of the sets in (2.5) holds as soon as R is sufficiently large. Indeed, suppose that F, F ∈ F and that z ∈ F maps to some point in F of modulus at least R. Since f (z) ∈ W 0 , we must in fact have z ∈ F , and additionally z ∈ ∞ F if R is sufficiently large. Now let ζ 0 and (ζ F ) F ∈F be defined as in the proof of Corollary 2.10. Let R be sufficiently large such that any point z ∈ W 0 of modulus at least R has dist W 0 cyl (ζ 0 , z) ≥ 3 max(C, 2π).
Let z be a point whose orbit is contained in F ∈F F , and furthermore |z| ≥ R. Then it follows that It follows by induction that dist W 0 cyl (ζ 0 , f n (z)) → ∞ uniformly in n, and the claim follows.
Exponentially bounded addresses. As noted above, the results on bounded addresses can be generalised to certain unbounded addresses. While we do not require this fact for this paper, we shall record it for future reference.
Definition 2.12 (Exponentially bounded addresses). Let ζ 0 ∈ W 0 be an arbitrary base point. For any fundamental domain F , let ζ F be the unique preimage of ζ 0 in F . We say that an infinite external address s is exponentially bounded if there exists a positive real number T with the following property. For all n ≥ 0, if ζ Fn ∈ ∞ F n , then If ξ is another base-point, then it follows from 2.4 that dist W 0 cyl (ζ F , ξ F ) is uniformly bounded (where defined) for all fundamental domains F . Thus it follows that the definition of exponentially bounded addresses is independent of the choice of base point ζ 0 .
Exponentially bounded addresses were defined previously for exponential maps [SZ03a] and cosine maps [RS08b]; it is easy to see that in these cases the definition agrees with ours. For these families, the class of exponentially bounded addresses s agrees precisely with those for which J 0 s = ∅. This is no longer true for general f , even when f has finite order of growth; see [ABR].
In [BK07,Corollary B'], it is shown that J 0 s = ∅ for a certain class of addresses, namely those whose orbits remain within finitely many tracts, and such that the "index" of the corresponding fundamental domains within each tract does not grow faster than an iterated exponential. It is easy to see that such addresses are exponentially bounded in our sense, but the converse is not the case. (For example, our definition allows for addresses taking values in fundamental domains that lie in infinitely many different tracts.) We now show that (c) and (d) of Theorem 2.5 can be extended to exponentially bounded addresses as follows.
Proposition 2.13 (Exponentially bounded addresses are realised). Let s be an exponentially bounded address. Then J 0 s = ∅. More precisely, there is a number R > 0, depending only on the base-point ζ 0 ∈ W 0 and T > 0, with the following property. If s is exponentially bounded for this choice of ζ 0 and T , then J 0 s contains an unbounded connected set X on which the iterates tend to infinity uniformly, and which contains a point of modulus R.
Conversely, the iterates of f tend to infinity uniformly on where the union is taken over all external addresses s as above.
Proof. We shall use an expansion estimate [RRRS11, Lemma 3.1], which is stronger than (2.4) at large distances. (Compare also [BK07, Lemma 3.3].) We will use this estimate in the following form, which follows easily from the version stated in [RRRS11]: There are constants C 1 , C 2 > 0 with the following property. If F is a fundamental domain and ζ 1 , ). Now fix ζ 0 and T > 0, and denote by S the set of all addresses that satisfy Definition 2.12 for these choices. We may assume without loss of generality that ζ 0 is chosen sufficiently large to ensure that ζ F ∈ ∞ F for every fundamental domain F . (To this end, we may need to increase T , but only by a finite amount according to the remark following Definition 2.12.) Define E(t) . . = exp(C 2 · t). By basic properties of exponential growth, there isR > 0 such that the following hold for all x ≥R/3.
). Indeed, this is trivial for n = 0, and if the claim holds for n, then Hence, by (2.8) and 2.9, ). The claim follows by induction. In particular, Since this applies to all points in the union (2.7), the second claim of the proposition follows. Now let us prove the first.
. Such a point exists because dist(ζ Fn , ζ 0 ) < E n (t), and ζ Fn can be connected to infinity within ∞ F n . For n ≥ 0, let Γ n be the union of cross-cuts from Lemma 2.7, applied to z n , and let A k n , for n, k ≥ 0, be defined precisely as in the proof of Theorem 2.8.
n ∩ Γ n = ∅, then the claim follows by choice of z n . In particular, this is always the case for k = 0.
So suppose that k ≥ 0 is such that the claim holds for all n ≥ 0, and that n is such that A k+1 n ∩ Γ n = ∅. Then, by definition, f (A k+1 n ) = A k n+1 . By the inductive hypothesis, the latter contains a point w with Then, by (2.8) and (2.9), In particular, the set k≥0 A k 0 has a connected component X containing a point with dist(A k n , ζ 0 ) ≤ 3t =R, and hence of modulus at most R. Since X is connected and unbounded, it also contains a point of modulus exactly R. The proof is complete.

Hyperbolic expansion and fundamental tails
For the remainder of the article, we shall specialise to the case where our transcendental entire function f has bounded postsingular set P(f ).
This section collects some fundamental preliminary material concerning these functions. We begin by noting a global expansion property away from the postsingular set, and then proceed to introduce the notation of fundamental tails, which will later be used to study the structure of the set of escaping points.
Recall that if Ω is a hyperbolic domain, we denote by ρ Ω the density of the hyperbolic metric on Ω.
Proposition 3.1 (Hyperbolic expansion). For every transcendental entire function f , ⊂ Ω, then f is strictly expanding at z in the hyperbolic metric of Ω. Moreover, this expansion factor tends to infinity as z → ∞ in V . That is, This result was proved, for a hyperbolic entire function f and a certain choice of P, in [Rem09, Lemma 5.1]. The same proof goes through whenever P intersects the unbounded connected component of C \ S(f ). This can always be ensured by adding a periodic orbit to P that intersects this component, which is sufficient for all our purposes. However, for completeness and future reference, we shall prove the result for general P, with a slightly simpler proof than that given in [Rem09]. To do so, we use the following simple fact.
Lemma 3.2 (Preimages in annuli). Let f be an entire transcendental function which is bounded on an unbounded connected set. Let z 1 , z 2 ∈ C. Then, for all C > 1, and all sufficiently large R, f −1 ({z 1 , z 2 }) contains a point of modulus between R/C and C · R.
Proof. Let C > 1 and set A . . = {z ∈ C : 1/C < |z| < C}. Suppose by contradiction that R n → ∞ is a sequence such that the functions omit both z 1 and z 2 . By Montel's theorem, this sequence of functions is normal, and hence converges locally uniformly, possibly after restricting to a subsequence. By assumption, lim sup min |z|=1 g n (z) < ∞, and hence the limit function is holomorphic. But this implies that f remains bounded on the circle of radius R n as n → ∞. Hence f is bounded by the maximum principle and hence constant by Liouville's theorem, a contradiction.
Proof of Proposition 3.1. The fact that #P(f ) ≥ 2 is well-known: Otherwise, f would be a self-covering of a punctured plane, and hence conformally conjugate to z → z d for some d. However, f is transcendental.
So Ω is indeed a hyperbolic domain. Since f : V → Ω is a covering map, and hence a local isometry with respect to the hyperbolic metrics of V and Ω, The open mapping theorem implies that every connected component of Since #P ≥ 2, and by Picard's theorem, C \ V = C \ f −1 (P) contains points of arbitrarily large modulus. Hence V Ω. By Pick's theorem, it follows that for all z ∈ V . This establishes the first claim.
Furthermore, since f ∈ B, there is an unbounded connected set on which f is bounded. (For example, the boundary of one of the tracts of f ). By Lemma 3.2, there is a sequence (c n ) n≥0 in Ω\V such that c n → ∞ and |c n+1 /c n | ≤ 2. (We can even ensure |c n+1 /c n | → 1 by letting the constant C in Lemma 3.2 tend to 1, but do not require this here.) Hence  Let Ω, P be as in Proposition 3.1, and suppose that U ⊂ Ω is open with f n (U ) ⊂ Ω for all n. Then dist(f n (z), P) → 0 uniformly on compact subsets of U .

Proof.
Note that (f n | U ) form a normal family by Montel's theorem, and hence U ⊂ F (f ). Therefore the first claim follows from the second: if dist(f n (z), P) → 0 for all z ∈ U , then this convergence is automatically uniform on compact subsets of U .
So suppose that z 0 ∈ Ω, and that there exist ε > 0 and an increasing sequence (n k ) ∞ k=0 such that dist(f n k (z 0 ), P) ≥ ε. We must show that z 0 ∈ J(f ). It follows from Lemma 3.2 that dist Ω (z, f −1 (P)) remains bounded as z → ∞ in Ω. Hence So we can connect f n k (z 0 ) to a point of f −1 (P) by a curve γ k of hyperbolic length at most δ in Ω. Pulling back γ k under f n k , we obtain a curve α k connecting z 0 to a point w k ∈ f −(n k +1) (P). By Proposition 3.1, each of the curves α j k = f n j (α k ), for j < k, has hyperbolic length at most δ. Hence these curves stay a uniform distance away from P, and again by Proposition 3.1, there is λ > 1 such that for all k, all j < k, and all z ∈ α j k .
So in fact α k has hyperbolic length at most δ · λ k , and w k → z 0 as k → ∞. converging to z 0 . But f n k+1 (w k ) ∈ P, and dist(f n k 1 (z 0 ), P) ≥ ε. Therefore the family of iterates of f is not equicontinuous at z 0 , and z 0 ∈ J(f ), as desired.
Remark. The result can also be proved by appealing to the classification of Fatou components.
Fundamental tails. Let f be an entire transcendental function with bounded postsingular set, and denote the unbounded connected component of C \ P(f ) by Ω. (In everything that follows, we could more generally let Ω be as in Proposition 3.1; i.e. the unbounded connected component of C \ P where P is a forward-invariant compact set containing P(f ). However, we shall not require this extra generality.) Let D and γ be as in Section 2. We may additionally assume that D is chosen sufficiently large to ensure that P(f ) ⊂ D and that ). Recall that fundamental domains are the connected components of the preimage of W 0 = C \ (D ∪ δ) under f . All concepts that follow depend a priori on this choice of fundamental domains, i.e. on the choice of the initial configuration consisting of D and δ. However, it turns out that this choice is not essential. (Compare Observation 4.12.) The postsingular set P(f ) is contained in D, and the image of any fundamental domain is contained in C \ D. Hence the closure of a fundamental domain does not intersect the postsingular set. It follows that for any fundamental domain F , any n ≥ 0, and any connected component τ of f −n (F ), τ is a Jordan domain on the Riemann sphere whose boundary contains ∞ and whose closure is mapped homeomorphically onto F by f n . Moreover, f k (z) → ∞ as z → ∞ in τ , for all k ≥ n + 1.  If τ is a fundamental tail of level n > 1, then f (τ ) is a fundamental tail of level n − 1.
Proof. Recall that the fundamental tails of level N are precisely the connected components of f −N (W 0 ). Hence they are pairwise disjoint, and the second claim follows from Lemma 2.1. The final claim holds by definition.
Lemma 3.6 (Fundamental domain associated to fundamental tail). Let τ be a fundamental tail of level n. Then there is a unique fundamental domain F such that F ∩ τ is unbounded. In fact, if n > 1, then F contains all sufficiently large points of τ .
Proof. We proceed by induction. The claim is trivial for n = 1. Now suppose that n > 1. By induction, there is a unique fundamental domain F 1 whose intersection with the fundamental tail f (τ ), of level n − 1, is unbounded.
Let A 1 be the unbounded connected component of F 1 \D, and let A 2 be the unbounded connected component of A 1 ∩ f (τ ) \ D. Then f (τ ) \ A 2 is bounded. Indeed, this is clear if n = 2, and otherwise follows from the inductive hypothesis. Recall It follows from the above that we can associate natural symbolic sequences to fundamental tails.
Definition 3.7 (Addresses of fundamental tails). Let τ be a fundamental tail of level n, and denote by F k (τ ) the unique fundamental domain whose intersection with the fundamental tail f k (τ ) is unbounded. We call the finite sequence s = F 0 (τ )F 1 (τ ) . . . F n−1 (τ ), of length n, the (finite) external address of τ .
Conversely, we can construct a fundamental tail having an arbitrary prescribed (finite) address by taking repeated pull-backs along the correct branches. Recall that a sequence s 1 , say of length n, is a prefix of another sequence s 2 of length m ≥ n if the first n entries of s 2 coincide with those of s 1 .
Definition and Lemma 3.8 (Tails at a given address). Let s = F 0 F 1 . . . be a finite or infinite sequence of fundamental domains having length at least n ≥ 1. Then there is a unique fundamental tail τ of level n having addresss . . = F 0 F 1 . . . F n−1 . We denote this fundamental tail by τ n (s). We also define the inverse branches Proof. The proof is by induction on the level n of the tails. For n = 1 and for every fundamental domain F 0 , the fundamental tail of level 1 and address F 0 is the fundamental domain F 0 itself, which is unique and has address F 0 by definition. Now let s be in the claim and let τ = τ n−1 (σ(s)) be the fundamental tail of level n−1 and address F 1 . . . F n−1 . This tail exists and is unique by the inductive hypothesis. Let R be sufficiently large, and let τ 1 be the unique unbounded connected component of τ \ D R (0). By Lemma 3.6, τ 1 is contained in the fundamental domain F 1 = F 0 (τ ) for sufficiently large R. Hence, if we additionally assume that R is greater than the radius of D, we have τ 1 ⊂ f (F 0 ) = W 0 . So there is a unique connected componentτ 1 of f −1 (τ 1 ) contained in F 0 , and a unique connected componentτ of f −1 (τ ) containingτ 1 . Then F 0 (τ ) = F 0 ; i.e., F 0 is the initial entry in the address of the fundamental tailτ , which hence has address s.
The following are immediate consequences of the preceding results and definitions.
Observation 3.9. Let τ be a fundamental tail of level n, and let s be the address of τ . Then the address of f (τ ) is σ(s), where σ denotes the shift map.
Suppose that τ 1 and τ 2 are fundamental tails of levels n 1 and n 2 , with n 1 ≥ n 2 . Let s 1 and s 2 be the addresses of τ 1 and τ 2 , respectively. Then τ 1 ∩ τ 2 is unbounded if and only if s 1 is a prefix of s 2 . In this case, if addditionally n 1 > n 2 , all sufficiently large points of τ 1 lie in τ 2 .

Filaments
Maintaining the same notation as in the previous section, we now define and study the central objects of this article: filaments. Recall that P(f ) is bounded; the main goal of this section is to show that, under this assumption, each of the sets J 0 s defined in Section 2 can be consistently extended to a larger -and, in a certain sense, maximal -set J s . The intersection of J s with the escaping set I(f ) forms the filament G s at address s.
As we shall see, each filament is an unbounded connected set of escaping points, the escaping set can be written as the union of filaments, and via their external addresses the collection of filaments is endowed with a natural combinatorial structure. Furthermore, the definition of filaments does not depend on the initial choices made in the construction of fundamental domains. Together these facts indicate that filaments can indeed be considered a natural generalisation of "hairs" or "rays".
For a fundamental domain F , recall that ∞ F denotes the unbounded connected component of F \ D. We extend this definition to fundamental tails as follows.
Definition 4.1 (Unbounded parts of tails). Let τ be a fundamental tail of level n. We define Observe that, if s is an external address and n ≥ 2, then ∞ τ n (s) is precisely the unbounded connected component of τ n (s) ∩ τ n−1 (s).
Definition 4.2 (Filaments). Let s be an (infinite) external address. We say that a point z ∈ C has external address s if z ∈ ∞ τ n+1 (s) for all sufficiently large n. We denote the set of all points z ∈ C having external address s by J s . The filament G s is defined to be G s . . = J s ∩ I(f ).
Remark. If z ∈ J s , then f n (z) belongs to ∞ F n for all sufficiently large n. In particular, z ∈ J(f ) by Lemma 2.6. Observe, however, that J s is not closed in general.
Note that other notions of points having external address s appear in the literature; for example, in [Rem07a], z is said to have address s = F 0 F 1 . . . if f n (z) ∈ F n for all n ≥ 0. The advantage of the above definition, in the context of postsingularly bounded functions, is that we shall see that every escaping point has an external address (Corollary 4.5), and that this address is in a certain sense independent of the initial choice of fundamental domains (Observation 4.12).

Lemma 4.3 (Properties of addresses and filaments).
Suppose that s is an external address and z ∈ J s . Then (a) z / ∈ Js for s =s.
, then w has an address in σ −1 (s), and every such address is realised by exactly one element of f −1 (z).
Proof. The first claim is trivial since fundamental tails of a given level are disjoint. Now By definition, z belongs to ∞ τ n (s) for all sufficiently large n; say for n ≥ n 0 . Let In particular, for n ≥ n 0 + 1, there is a fundamental tail ϑ n such that f (ϑ n ) = τ n−1 (s) and w ∈ ∞ ϑ n . Recall that ∞ τ n (s) ⊂ τ n−1 (s), and that the intersection ∞ ϑ n+1 ∩ ϑ n is non-empty (since it contains w). Hence ∞ ϑ n+1 ⊂ ϑ n for all n ≥ n 0 + 1. That is, ϑ n+1 tends to infinity within ϑ n . So if F is the fundamental domain whose intersection with ϑ n 0 +1 is unbounded, then ϑ n tends to infinity in F for n ≥ n 0 + 1. It follows that ϑ n = τ n (F s). In particular, w has address F s ∈ σ −1 (s).
Conversely, lets ∈ σ −1 (s); that is,s = F s for some fundamental domain F . Then the fundamental tail ϑ n 0 +1 of level n 0 + 1 associated to the addresss . . = F s is a component of f −1 (τ n 0 ). Hence there is a unique element w of f −1 (z) in ϑ n 0 +1 , and w has addresss as above.
The final claim follows from the previous two.
The following proposition establishes a connection between the sets J s defined in Definition 4.2 and the sets J 0 s studied in Section 2.
Proof. This is essentially the content of the second paragraph of [Rem07a, Proof of Theorem 1.1]. Since that proof is somewhat concise, and we are using a different terminology, we provide the details. For n ≥ 1, let ϑ n denote the fundamental tail of level n containing z. By assumption, f n−1 (z) ∈ ∞ F n−1 , and hence z ∈ ∞ ϑ n . We claim that ∞ ϑ n+1 ⊂ ϑ n ; that is, ϑ n+1 tends to infinity within ϑ n . Indeed, Inductively, ϑ n tends to infinity within F 0 = ϑ 1 for each n ≥ 0. Applying this fact to f k (z), we see likewise that f k (ϑ n ) tends to infinity within F k for n ≥ k. Thus, for any n ≥ 1, we conclude that ϑ n = τ n (s), and hence z ∈ ∞ ϑ n = ∞ τ n (s), as desired.
There is a number R > 0 with the following property. If z ∈ J(f ) is such that |f n (z)| ≥ R for all sufficiently large n, then z has an external address s. (c) Every escaping point z ∈ I(f ) has an external address s, and hence belongs to a filament G s .
Proof. The "only if" direction of (a) is immediate from Definition 4.2. On the other hand, if z has the stated property, then f n (z) has address s by Proposition 4.4, and hence z also has an external address by Lemma 4.3.
To prove (b), let F be the set of fundamental domains that intersect D. Recall that F \ ∞ F is bounded for all F ∈ F, and that F is finite by Lemma 2.1. Now fix Suppose that |f n (z)| ≥ R for all n ≥ n 0 . Then, for all n ≥ n 0 , f n (z) belongs to some fundamental domain F , and by choice of R, it must belong to ∞ F . Now z has an address by (a).
Finally, (c) is an immediate consequence of (b).
Remark 4.6. Under the hypotheses of Proposition 4.4, f n (z) belongs to ∞ F n for all n. However, it may well be that z belongs to a bounded component of τ k (s) \ D for all sufficiently large k and similarly for all points on the forward orbit of z.
Indeed, typically when I(f ) does not consist of hairs, exactly this is the case for many escaping points, so Proposition 4.4 is not at all trivial, and uses the boundedness of the postsingular set in an essential manner. No analogue thereof is currently known for functions with unbounded postsingular set, and in particular in this setting there is no canonical way of associating external addresses to arbitrary escaping points as in Corollary 4.5. This is a major challenge in showing the unboundedness of components of the escaping set. (Compare Corollary 4.11 below.) Connectedness of filaments and uniform escape to infinity. Thanks to Proposition 4.4, we can use the results of Section 2 to study filaments. We use the following definition from [Rem16]. Remark. While µ(z) is a union of sets on which the iterates tend to infinity uniformly, typically f n will not tend to infinity uniformly on µ(z).
Lemma 4.8 (Filaments and uniform escape). Let s be an external address, and suppose that z ∈ G s . Let A z be a connected set and suppose that there is N ≥ 0 such that f n (A) ∩ D = ∅ for n ≥ N . (In particular, this is the case when the iterates of f tend to infinity uniformly on A.) Then is a connected component of the set on the right-hand side, z ∈ A ∩ ∞ τ n (s), and A is connected. Hence A ⊂ ∞ τ n (s) for n ≥ n 0 , and all points in A have address s.
We remark that a filament G s may contain uncountably many different escaping composants. Moreover, it is possible for µ(z) to consist of a single point. (See [Rem16, Theorem 1.6].) However, by the results of Section 2, G s contains a distinguished uniformly escaping composant, namely the one consisting of those points for which the set A can be taken to be unbounded.
Definition and Lemma 4.9 (The core of a filament). Let s be an infinite external address. Let X be the collection of all closed, unbounded, connected sets X ⊂ G s on which the iterates of f tend to infinity uniformly. Then no element of X separates the plane. Furthermore, X is linearly ordered by inclusion; i.e., if X 1 , X 2 ∈ X then X 1 ⊂ X 2 or X 2 ⊂ X 1 .
In particular, if J s = ∅, then the union µ s . . = X ⊂ G s = ∅ satisfies µ s = µ(z) for all z ∈ µ s . We call µ s the core of the filament G s .
Proof. If X ∈ X , then there is n 0 ≥ 1 such that X ⊂ ∞ τ n (s) for n ≥ n 0 . In particular, f n (X) ⊂ J 0 σ n (s) for n ≥ n 0 . Let n ≥ n 0 . Since f n : τ n (s) → W 0 is a conformal isomorphism, and X is unbounded, it follows that f n (X) is unbounded for all n. By Lemma 2.6 we know that f n (X), and hence X, does not separate the plane.
Furthermore, if X 1 , X 2 ∈ X , then we can choose n 0 sufficiently large so that the above holds for both sets. It follows from part (b) of Theorem 2.5 that one of f n 0 (X 1 ) and f n 0 (X 2 ) is contained in the other, and the same holds for X 1 and X 2 .
By part (a) of Theorem 2.5, together with Proposition 4.4, the set µ s (as defined in the statement of the lemma) is non-empty. By the fact we just proved, µ s is a nested union of connected sets, and hence itself connected. Finally suppose that z ∈ µ s ; so z ∈ X z for some X z ∈ X . Since X is linearly ordered by inclusion, we have Conversely, if the iterates of f tend to infinity uniformly on the connected set A z, then they do so also on the closed, unbounded and connected set A ∪ X z , which is contained in G s by Lemma 4.8. Hence A ⊂ µ s , and we have proved µ(z) = µ s , as desired. Then µ s is dense in J s and G s . In particular, both of these sets are connected and unbounded.
Proof. Let z ∈ J s . By Corollary 4.5, there is n 0 such that f n (z) ∈ J 0 σ n (s) for n ≥ n 0 . Let X n be the unbounded connected subset of µ σ n (s) whose existence is guaranteed by Theorem 2.5. Recall that X n can be connected to f n (z) by a curve γ n ⊂ C \ D of cylindrical length at most 4π such that γ n is homotopic to a curve in W 0 . In particular, the pullbackγ n of γ n along the orbit of z connects z to the setX n . . = f −n s (X n ) ⊂ µ(s). The density ρ Ω of the hyperbolic metric on Ω tends to zero like 1/|z||log z|. (Recall that Ω is the unbounded connected component of C \ P(f ).) Therefore the hyperbolic length Ω (γ n ) is also uniformly bounded, independently of n. It follows by Proposition 3.1 that Ω (γ n ) → 0 as n → ∞. Hence dist Ω (z,X n ) → 0, and z ∈ µ s , as claimed.
Recall that the (relative) closure of a connected set is again connected. So J s and G s are connected and unbounded, as the dense subset µ s has these properties.
We note that the above result (and its proof) is essentially a reformulation of the main argument in the proof of the main theorem of [Rem07a], which we recover as follows. Proof. Let z ∈ I(f ). By Corollary 4.5, z ∈ G s for some external address s, and by Proposition 4.10, G s is an unbounded connected subset of I(f ).
Independence of filaments from the construction. Note that the definition of addresses, and hence of filaments, depends a priori on the choice of fundamental domains, and hence on the domain W 0 (i.e., on the choice of the disc D and the curve δ). However, if filaments are to be considered canonical objects associated to f , then this dependence should not be essential. We briefly discuss why this is indeed the case.
Suppose thatŴ 0 is a different choice of base domain, and thatτ is a fundamental tail of level at least 2, using this alternative initial configuration.
Then it is easy to see that there is a unique fundamental domain F for the original domain W 0 such that all sufficiently large points ofτ lie in F . Indeed, if z ∈τ is sufficiently large, then f (z), f 2 (z) / ∈ D, and in particular f (z) / ∈ δ. So f (z) ∈ W 0 , and the claim is established.
In particular, ifŝ is an external address with respect toŴ 0 , then we can associate toŝ an address s = F 0 F 1 . . . with respect to W 0 . Here for each n ≥ 0, F n is the fundamental domain associated, in the above manner, to the fundamental tailsτ k (σ n (ŝ)), for k ≥ 2. (Observe that F n is independent of k.) It is easy to see that, in turn,ŝ can be obtained from s using the same procedure in the opposite direction, so that the correspondence s →ŝ is a bijection between external addresses defined using the original configuration W 0 , and those defined using the modified one. The following observation is an easy consequence of this construction.
Observation 4.12 (Filaments are independent of the initial configuration). Let s and s correspond to each other as in the above construction; let G s be the filament obtained by using preimages of W 0 in the construction, and letĜŝ be the corresponding filament according to the choiceŴ 0 . Then G s =Ĝŝ.
In other words, the collection G = {G s } of subsets of J(f ) is independent of the initial choice of W 0 .
We remark that the same is not true for the set J s . Indeed, suppose that z is a nonescaping point that has an external address for one choice of W 0 . Then we can choose a different initial configurationW 0 , where the initial discD is chosen sufficiently large to ensure that z entersD infinitely many times. Clearly z does not have an external address with respect to this configuration.
In similar fashion, we see that filaments are preserved under iteration. Indeed, let n ≥ 1. Suppose thatD is a disc centred around zero such that f n−1 (D) ⊂D, and that δ ⊂ δ connectsD to ∞ outside ofD. ThenŴ 0 . . = C \ (cl(D) ∪δ) is a valid initial configuration for f n . Every fundamental domain of f n (with respect toŴ 0 ) is contained in a unique fundamental tail of level n for f . Conversely, every fundamental tail of f contains exactly one fundamental domain of f n . Hence there is a natural bijection between the fundamental domains of f n and finite external addresses of length n for f , and thus between external addresses of f n and those of f . In particular, we obtain the following.
Observation 4.13 (Filaments and iterates). Every filament of f is a filament of f n , for n ≥ 1, and vice versa.
Cyclic order of addresses and filaments. There is a natural cyclic order on the set of fundamental tails of any given level, and in particular on the set of fundamental domains of f : if A, B, C are fundamental tails, then A ≺ B ≺ C means that B tends to infinity between A and C in positive orientation. (See Section 13 for background on the cyclic order near infinity.) We can also define a cyclic order on the collection of filaments, by choosing for each a closed, connected, unbounded set on which the iterates tend to infinity, as in Proposition 4.10, and considering the cyclic order of these sets.
Recall that the function f acts in a natural way on filaments, and maps fundamental tails of level n + 1 to fundamental tails of level n. By the remark above, this action locally preserves cyclic order, in the following sense. Let A ≺ B ≺ C be either filaments or fundamental tails of some fixed level n > 1. If the addresses of A, B and C all have the same initial entry, then We can also define a "lexicographical" cyclic order on (finite or infinite) external addresses. To do so, we use the curve δ to convert the cylic order on fundamental domains to a linear order in the usual sense, setting F <F if and only if δ ≺ F ≺F . This linear order gives rise to a lexicographic order < on external addresses in the usual sense. The cyclic order on addresses is then defined by s 1 ≺ s 2 ≺ s 3 if and only if s 1 < s 2 < s 3 , s 2 < s 3 < s 1 , or s 3 < s 1 < s 2 .
It follows from what was written above that this cyclic order on addresses agrees with the cyclic order of the associated fundamental tails or filaments.
Disjoint-type addresses. To conclude this section, we discuss a particularly wellbehaved type of filament. Note that, in general, the points in the closure of a filament G s need not belong to J s . Indeed, this is the case for those filaments of greatest interest to us, namely those accumulating on a periodic point whose orbit does not lie outside of D. We show now that this cannot occur when the address s (or a forward iterate thereof) contains only fundamental domains that do not intersect D.
Lemma 4.14 ((Eventually) disjoint-type addresses). Let s = F 0 F 1 . . . be an external address, and suppose that there is n 0 ≥ 0 such that F n does not intersect ∂D for n ≥ n 0 . Then G s = J s = f −n s (J 0 σ n (s) ) ⊂ τ n (s) for n ≥ n 0 .
Proof. By assumption, we have F n ⊂ W 0 , and in particular ∞ F n = F n , for n ≥ n 0 . Thus τ n+1 (s) ⊂ τ n (s) for n ≥ n 0 (where we use the convention that τ 0 (s) = W 0 for convenience). In particular, ∞ τ n+1 (s) = τ n+1 (s), and by definition. Furthermore, (where the first equality holds by Proposition 4.10). This completes the proof.
Remark 4.15 (Disjoint-type addresses). If n 0 = 0, then we say that s is of disjoint type (since τ n (s) and τ n (s) have disjoint boundaries for n = n ). For a disjoint-type function (i.e., one that is hyperbolic with connected Fatou set), all addresses are of disjoint-type, given a suitable choice of fundamental domains. In this case, every component of the Julia set J(f ) is one of the sets J s = J 0 s , and the closure J s ∪ {∞} inĈ is called a Julia continuum. Compare [Rem16]. Suppose that, additionally, G s is a hair (in the sense of Definition 5.1 below). Then it follows, using [Rem16, Corollary 5.6], that either G s = J s , and J s is an arc to ∞ on which the iterates of f tend to infinity uniformly, or J s \ G s contains a single non-escaping point z 0 ; here G s is an arc connecting z 0 to ∞. (For functions satisfying a head-start condition, this follows already from the results of [RRRS11], without the need for the results of [Rem16].) In fact, it can be shown that, whenever s is of disjoint type, the set J s is homeomorphic to the component of the Julia set of a suitable disjoint-type function, with escaping points corresponding to escaping points. (Compare also Theorem 7.2 below.) Hence the above observation remains true for disjoint-type addresses, even if f itself is not of disjoint type. So we may think of filaments at disjoint-type addresses as always "landing". When s is bounded, we show below that this is true in a precise sense (see Proposition 6.5 (d)).
However, when G s is not a hair, it may happen that J s contains a dense or uncountable set of non-escaping points, even when f and hence all addresses are of disjoint type [Rem16, Theorem 2.3]. In this article, we only consider landing properties for filaments at bounded external addresses, so these subtleties will not become relevant.

Hairs and filaments
We shall now discuss the relationship between hairs and filaments. The term "hair" was coined by Devaney in the 1980s (see [Dev84,p. 168]), and is commonly used in an informal manner to refer to dynamically natural curves in Julia sets of transcendental entire functions. We use the following convention. (See Remark 5.6 below for comparison with some other definitions in the literature.) Remark 2. The second part of condition (a) is essential. Indeed, by [Rem16, Theorem 2.10], there exists a postsingularly bounded entire function having a filament which is an arc connecting a finite endpoint to infinity, but such that the iterates on this arc do not tend to infinity uniformly. On the other hand, the second part of condition (b) can be shown to be inessential. That is, suppose γ : (0, ∞] → G s ∪ {∞} is a continuous bijection with γ(∞) = ∞. Then, possibly after reversing the orientation of γ on (0, ∞), for all t > 0 we have f n | γ([t 0 ,∞)) → ∞ uniformly. We shall not provide the (somewhat lengthy) proof, as we do not require this fact in our paper.
(a) G s is a hair if and only if G σ(s) is a hair. Up to suitable reparameterisation, the corresponding functions γ andγ satisfyγ(t) = f (γ(t)). (b) If s is a bounded external address, then f n | Gs does not tend to infinity uniformly.
In particular, if s is bounded, then G s is a hair if and only if condition (b) of Definition 5.1 holds.
Proof. To prove the first claim, note that f : G s → G σ(s) is a continuous bijection. Furthermore, for every n ≥ 2, f | τn(s) extends to a homeomorphism τ n (s) ∪ {∞} → τ n−1 (σ(s)) ∪ {∞}. If G s is a hair, defineγ as in the claim. Clearly it is only necessary to show thatγ(t) → ∞ as t → ∞, which follows from the above fact since γ| [1/2,1) ⊂ τ n for sufficiently large n. For the converse direction, we define γ(t) . . = (f | τn(s) ) −1 (γ(t)), where n is sufficiently large depending on t, and proceed analogously. Now suppose that s is bounded, and let R > 0 be as in Corollary 2.10. Then, for all n ≥ 0, the filament f n (G s ) = G σ n (s) contains a point of modulus at most R. Hence the iterates of f do not tend to infinity uniformly on G s .
The following is an alternative formulation of Definition 5.1, and allows us to connect the notion with our definition of criniferous functions.

Proposition 5.3 (Characterisation of hairs). A filament G s of f is a hair if and only
if, for every z ∈ G s , there is an arc connecting z to ∞ on which f n → ∞ uniformly.
Proof. If G s is a hair, then the stated condition holds by definition.
So now suppose that every point z ∈ G s can be connected to infinity by an arc γ z on which the iterates tend to infinity uniformly. So γ z ⊂ µ(z) ⊂ µ s ⊂ G s by Lemma 4.8. By Lemma 4.9, the arcs γ z are linearly ordered by inclusion, and the arc γ z is unique.
Let (x n ) ∞ n=0 be a countable dense subset of G s . Set n 0 = 0 and define n k+1 inductively as the minimal value of n for which x n does not lie on the arc γ xn k . (If no such n exists, then G s = γ xn k satisfies Definition 5.1 (a) and we are done.) We set y k . . = x n k . Then the union of the arcs γ y k is a single continuous injective curve γ : (0, ∞) → G s , which can be parameterised such that γ([1/k, ∞)) = γ y k for all k.
First suppose that γ(t) has a limit z 0 ∈ G s as t → 0. Then γ z 0 = γ ∪ {z 0 } = γ = G s , since γ is dense in G s . Hence G s satisfies Definition 5.1 (a).
Otherwise, γ (0, ∞) ⊂ γ z for all z ∈ G s . Hence we see that γ z ⊂ γ y k for sufficiently large k; it follows that γ is surjective, and Definition 5.1 (b) holds.

Corollary 5.4 (Filaments and criniferous functions). A transcendental entire function f with bounded postsingular set is criniferous if and only if every filament is a hair.
Proof. Recall that every escaping point of f belongs to some filament. Hence the claim is immediate from Proposition 5.3.
Observe that we now have two apparently different definitions of periodic hairs, namely Definition 1.3, and the case of a periodic filament (i.e., a filament G s at a periodic address s) that is a hair. We conclude the section by showing that the two coincide.
Proposition 5.5 (Periodic hairs). Every periodic hair in the sense of Definition 1.3 is a periodic filament, and this filament is a hair in the sense of Definition 5.1. Conversely, any periodic filament that is a hair has a parameterisation as a periodic hair in the sense of Definition 1.3.
Proof. Suppose that γ is a periodic hair. Then, for every t ∈ R, the iterates f n tend to infinity uniformly on γ [t, ∞) . Hence, by Lemma 4.8, γ is contained in a filament G s , whose address s is necessarily periodic. Let X be as in Lemma 4.9. If X ∈ X , then f n (X) ⊂ γ [0, ∞) for sufficiently large n, and hence X ⊂ γ [−n, ∞) . Thus we conclude that γ = µ s .
Let R be as in part (d) of Theorem 2.5, applied to the set of fundamental domains occurring in s = F 0 F 1 F 2 . . . . It follows that there is t such that γ((−∞, t)) contains only points of modulus less than R. Indeed, by choice of R there is n 1 such that f n (z) / ∈ γ([0, 1]) for n ≥ n 1 , whenever z ∈ G s with |z| ≥ R; then we may take t = 1 − n 1 . Let z ∈ G s , and let n 0 be such that f n (z) ∈ F n and |f n (z)| > R for n ≥ n 0 . We may assume that n 0 is a multiple of the period of s. Recall that f n 0 (z) ∈ G s is in the closure of γ by Proposition 4.10. Since γ((−∞, t)) contains no points of modulus greater than R, it follows that f n 0 (z) ∈ γ([t, ∞)). Thus z ∈ γ([t − n 0 , ∞)). We have proved that G s = γ, and clearly G s is a hair in the sense of Definition 5.1. Now suppose that s is periodic and that G s is a hair. Letγ : (0, ∞) → G s be a parameterisation of G s as in part (b) of Definition 5.1. Consider the point z 0 . . =γ(1), and its image z 1 = f (z 0 ), then z 1 =γ(t) for some t. It follows easily that t > 1, and that the pieceγ (1, t) is disjoint from its forward and backward images in G s . We may reparameteriseγ to a curve γ : R → G s such that γ([0, 1]) corresponds toγ([1, t]), and such that γ(t + 1) = f (γ(t)) for all t. Then γ is a periodic hair in the sense of Definition 1.3.
Remark 5.6 (On the concept of hairs). The notion of a "dynamic ray" of an entire function is given in [RRRS11, Definition 2.2]. By the same reasoning as in Observation 5.2, in our setting of postsingularly bounded functions this definition can be phrased as follows: a "dynamic ray" of a postsingularly bounded entire function is a maximal curve in the escaping set satisfying (b) of Definition 5.1. Hence every dynamic ray is contained in the core of a filament, and a filament G s is a dynamic ray if and only if it satisfies Definition 5.1 (b). In particular, for a bounded address s, G s is a dynamic ray if and only if it is a hair. Moreover, for criniferous functions -and in particular the class of functions for which hairs are constructed in [RRRS11] -every dynamic ray in the sense of [RRRS11, Definition 2.2] is either a hair in the sense of Definition 5.1, or becomes such upon the addition of a finite escaping endpoint.
In general, [RRRS11] leaves open the possibility that a filament contains a hair as a proper subset. For example, it follows from [Rem16, Theorem 2.5] that there is an entire function f with bounded postsingular set (and indeed of disjoint type), having a bounded-address filament G s with the following properties. The set G s \ G s consists of a single point z 0 with bounded orbit, andĜ s is homeomorphic to a sin(1/x) continuum, with the starting point of the accumulating curve corresponding to ∞, and one of the endpoints of the limiting interval situated at z 0 . Then the accumulating curve itself is a "dynamic ray", but this ray does not include all points of G s .
Here, we restrict only to consider cases where the entire filament G s is a hair, and hence leave open the question of whether proper subsets of filaments should be considered "hairs" or not.
Also note that [RRS10, p. 740] defines a notion of Devaney hairs; for postsingularly bounded functions such a Devaney hair is a curve γ as in (b) of Definition 5.1, with the addition of a finite, not necesarily escaping, endpoint. In particular, if G s is a hair, then G s contains many Devaney hairs in the sense of [RRS10], linearly ordered by inclusion. Conversely, any Devaney hair is either contained in a filament, or consists of a filament together with a finite landing point.

Accumulation sets and landing properties of bounded-address filaments
We now study the accumulation behaviour of filaments at bounded external addresses. (The restriction to bounded addresses is due to the phenomena discussed at the end of Remark 4.15.) In the case where the filament in question is a hair, it is clear that the "accumulation set" of this filament should be the set of limit points of the curve γ(t) as t → 0, where γ is the curve from Definition 5.1. For periodic hairs, one can see easily that this is equivalent to fixing a base point on this hair, considering its successive preimages along γ, and studying the accumulation set of this sequence. This motivates the following definition.
Definition 6.1 (Accumulation sets of bounded addresses). Let s be a bounded external address, and let ζ ∈ W 0 .
For n ≥ 1, set ζ n . . = ζ n (s) . . = f −n s (ζ) ∈ τ n (s). (Recall from Definition 3.8 that f −n s = (f n | τn(s) ) −1 .) Then the accumulation set Λ(s, ζ) of s with respect to ζ is defined to be the accumulation set (inĈ) of the sequence (ζ n ) ∞ n=1 . Note that this is an abstract definition of the accumulation set associated to an address; it uses only the notion of fundamental tails, and does not require the definition of filaments or their properties. That this is a natural concept may not be clear a priori, but should become apparent through the results proved in this section. We begin by verifying that Λ(s, ζ) is independent of ζ.
This follows from the following well-known fact about the shrinking of univalent preimages. (Compare also [Lyu83, Proposition 3].) Recall that Ω is the unique unbounded component of C \ P (f ).

Lemma 6.2 (Euclidean shrinking). Suppose that V
Ω is a bounded Jordan domain. Then, for any ε > 0 and any compact set K ⊂ C, there exists N ε with the following property. For n ≥ N ε , every connected component of f −n (V ) that intersects K has Euclidean diameter at most ε.
Proof. Suppose, by contradiction, that there is a sequence (V k ) ∞ k=0 of n k -th preimages of V , with n k → ∞, with V k ∩ K = ∅ and inf k diam(V k ) > 0. LetṼ Ω be a slightly larger Jordan domain than V with V ⊂Ṽ , and letṼ k be the component of f −n k (Ṽ ) containing V n k . Then f n k :Ṽ k →Ṽ is a covering map and hence a conformal isomorphism, whose inverse ϕ k . . = (f |Ṽ k ) −1 :Ṽ →Ṽ k maps V to V k .
By assumption, there is a sequence (z k ) ∞ k=0 with z k ∈ K ∩ V k . By Koebe's distortion theorem,Ṽ k contains a round disc around z k whose diameter is comparable to that of V k . By assumption, the latter is bounded from below. Hence, if U is a sufficiently small disc centred at a limit point of the sequence (z k ), then U is contained in infinitely manỹ V n k . It follows that f n (U ) ⊂ Ω for all n ≥ 0, and f n (U ) ⊂Ṽ for infinitely many n. This contradicts Corollary 3.3.

Corollary 6.3 (Spherical shrinking). Suppose that V
Ω is a bounded Jordan domain. Then, for any ε > 0 there exists N such that for n ≥ N every connected component of f −n (V ) has spherical diameter at most ε.
Proof. Let K ⊂ C be a compact set whose complement has spherical diameter less than ε. Since the spherical and Euclidean metrics are comparable on any compact subset of the plane, there is ε 1 such that any set of Euclidean diameter at most ε 1 that intersect K has spherical diameter at most ε. Let N = N ε 1 be as in Lemma 6.2, let n ≥ N , and let X be a connected component of f −n (V ). Then either X ∩ K = ∅, and hence X has spherical diameter less than ε, or X ∩ K = ∅ and Lemma 6.2 applies. In the latter case, diam X ≤ ε 1 , and hence X also has spherical diameter at most ε by choice of ε 1 .
We write Λ(G s ) . . = Λ(s) . . = Λ(s, ζ), and call Λ(G s ) the accumulation set of the filament G s . The filament G s is said to land at a point z 0 ∈Ĉ if Λ(G s ) = {z 0 }.
Proof. Let V be a bounded Jordan domain in W 0 containing both ζ 1 and ζ 2 . For j ∈ {1, 2} and n ≥ 1, write ζ n j . . = f −n s (ζ j ) ∈ τ n . . = τ n (s). Since f n : τ n → W 0 is univalent, for any n the points ζ 1 n and ζ 2 n belong to the same connected component of f −n (V ). By Corollary 6.3, the spherical diameter of this component tends to zero as n → ∞. Hence the sequences (ζ 1 n ) ∞ n=1 and (ζ 2 n ) ∞ n=1 have the same accumulation set inĈ. The following establishes a number of fundamental properties of accumulation sets.  (c) If J s \ G s = ∅, then this set has a unique element z 0 , and G s lands at z 0 , which has bounded orbit. (d) If s is of disjoint type, then G s lands at a point z 0 ∈ J s having bounded orbit. (e) Let U be a neighbourhood of a point z 0 ∈ Λ(s). Then f n does not tend to infinity uniformly on U ∩ G s . (f ) Let K ⊂Ĝ s be compact with K ∩ Λ(s) = ∅. Then f n → ∞ uniformly on K ∩ C, and in particular K ∩ C ⊂ J s .
Before proving these facts, we observe a few consequences. Firstly, we see that we can characterise the accumulation set purely in terms of G s as a subset of the escaping set, justifying the notation Λ(G s ) = Λ(s). Corollary 6.6 (Topological landing criterion). Let G s be a filament at a bounded address s. Then the accumulation set of G s consists precisely of those points inĜ s having a neighbourhood U such that f n does not tend to infinity uniformly on U ∩ G s .
In particular, G s lands at a point z 0 ∈ C if and only ifĜ s = G s ∪ {z 0 , ∞} and if furthermore, for every neighbourhood U of z 0 inĈ, the iterates of f tend to infinity uniformly on G s \ U .
Proof of Corollary 6.6, using Proposition 6.5. The first claim is a direct consequence of Proposition 6.5 (e) and (f). The second claim follows from the first (together with the fact that the accumulation set of G s is nonempty).
In particular, it follows that for hairs our definition of the accumulation set agrees with the usual one. Recall by Observation 5.2 that if a bounded-address filament is a hair, it must satisfy part (b) of Definition 5.1.
Corollary 6.7 (Accumulation sets of hairs). Suppose that a filament G s at bounded address is a hair, and let γ : (0, ∞) → G s be the continuous bijection from Definition 5.1. Then Λ(G s ) is precisely the accumulation set of γ(t) as t → 0.
This follows from a well-known argument that we sketch as follows; compare [Rem06b, Lemma 5.1] for the case of exponential maps. For every t > 0, there are pieces of other hairs of f accumulating uniformly from above and below on γ [t, ∞) . (For example, this follows from [Rem16, Proposition 8.1] via [Rem09, Theorem 1.1].) Thus, in order to accumulate on γ(t 0 ), the curve γ must also accumulate on γ([t, t 0 ]); letting t → 0, the claim is established.
Let t 0 ∈ (0, ∞] be the supremum over all possible choices of t 0 , and let (t n ) ∞ n=0 be a decreasing sequence with t n → 0. Then A n . . = γ([t n , t 0 ]) is a compact and nowhere dense subset of Λ(γ) for all n. By Baire's theorem, Λ = Λ(γ) \ A n is dense in Λ(γ), as claimed. This completes the proof.
Remark 1. The proof shows that a hair G s either lands, or otherwise a generic point in Λ(G s ) belongs to C \ G s . It is plausible that this is true without the assumption that G s is a hair, with a similar proof. This would simplify the characterisation in Corollary 6.6 as follows: A filament G s at a bounded address s lands at a finite point z 0 ∈ C if and only if G s \ G s = {z 0 }.
Remark 2. For periodic addresses, or for addresses satisfying a head-start condition as in [RRRS11], the proof of Corollary 6.7 is considerably simpler. Indeed, in this setting it is easy to see directly that the iterates of f do not tend to infinity uniformly on any neighbourhood of any point of Λ(γ).
Finally, we observe that most periodic rays land and most periodic points are landing points. Compare also [BK07,BF15,Ben16].
Corollary 6.8 (Most periodic filaments land). Let p ≥ 1. Then, for all but finitely many periodic addresses s of period p, the filament G s lands at a periodic point z ∈ J s of period p. Conversely, for all but finitely many periodic points z of period p, the point z is repelling and there is a periodic address s of period p such that z ∈ J s ; in particular, G s lands at z.
Proof. Passing to an iterate, we can assume that p = 1. (Recall that the sets J s are pairwise disjoint by Lemma 4.3 (a).) Only finitely many fundamental domains F intersect D, and hence all but finitely many fixed addresses of f are of disjoint type. Hence the first claim follows from Proposition 6.5 (d). Similarly, all but finitely many fixed points of f are contained in W 0 , and hence have a fixed external address. So the second claim is a consequence of Proposition 6.5 (c).
The remainder of the section is dedicated to establishing Proposition 6.5. We shall do so by applying Corollary 6.3 to a suitable large Jordan domain V , depending on the collection of fundamental domains involved. The following technical lemma collects the properties that we require of V . Lemma 6.9 (Domains for bounded-address filaments). Let ζ belong to an unbounded connected component of W 0 ∩ f −1 (D) and let R > 0. Let F be any finite collection of fundamental domains of f . Then there is a Jordan domain V Ω with the following properties.
(ii) For all F ∈ F, the unique preimage ζ F of ζ in F also belongs to V .
(iii) For all F ∈ F, there is a connected component A F of V ∩ F containing ζ F as well as all points of F having modulus at most R.
Proof. See Figure 3. Write F = {F 1 , . . . , F n }. Let γ be a Jordan curve in C \ D that intersects the arc δ in exactly one point. We may assume that γ is chosen so large that γ surrounds ζ, f (D) and the set {f (z) : |z| ≤ R}.
For each i, set γ i . . = f −1 (γ) ∩ F i and ζ i . . = ζ F i . Then γ i is a cross-cut of F i that separates ζ i , F i ∩ D and all points of modulus at most R in F i from ∞ in F i . In particular, γ i belongs to the unbounded connected component of Figure 3. A domain V as described in Lemma 6.9. The domain V is shaded. For clarity, we include only the fundamental domains F i that comprise the collectionF ; each of these is contained in a tract of f and is adjacent to other fundamental domains, which are not shown.
Let X i be the closure of the bounded component of F i \ γ i , and let Y be the union of these components. Observe that different X i may intersect when the corresponding fundamental domains are adjacent. In this case, the corresponding two X i intersect precisely in a preimage component of the piece of δ that connects D to γ. Let Y 1 , . . . , Y k be the k ≤ n connected components of X, and set Γ j . . = Y j ∩ f −1 (γ). That is, Γ j is the union of finitely many γ i , together with their endpoints.
Then each Y k is a closed Jordan domain in Ω \ δ, and Γ j is an arc of ∂Y k . (Recall that the closure of a fundamental domain does not meet δ.) Let Y be obtained from Y by adding, for each j ≤ k, an arc β j in W 0 \ X joining ζ to a point of Γ j ; this is possible by the assumption on ζ. We may assume that two different β j intersect only at ζ.
Then Y is a compact and full set, and we may let V be a Jordan domain containing Y . The point ζ and all ζ i belong to V by construction. Moreover, each Y j is connected, and hence belongs to a single connected component of V ∩ ( i F i ). Finally, the union of all β j and all Γ j is connected by construction, is contained in C \ D and intersects the unbounded connected component of F i \D for all i. This completes the construction.
The domain V from Lemma 6.9 allows us to study the accumulation sets of filaments. The following lemma is crucial not only in our study of accumulation sets, but also for the proofs of our main theorems. The key idea is that, in order to investigate the landing of a given filament G s , we can study a certain chain of simply connected domains (obtained as iterated preimages of the domain V from Lemma 6.9) whose diameters shrink to zero. Lemma 6.10 (Preimage domains). Let F be a finite collection of fundamental domains of f , and assume that F contains every fundamental domain F with F ∩ D = ∅. Let ζ, R and V be as in Lemma 6.9. If R was chosen sufficiently large (depending only on F), then the following holds.
Let s = F 0 F 1 . . . be any external address. For n ≥ 1, set ζ n (s) . . = f −n s (ζ) ∈ τ n . . = τ n (s). Also let V n = V n (s) be the unique component of f −n (V ) containing ζ n (s). Then the following hold for all n ≥ 1.
(1) The spherical diameter of V n (s) tends to 0 as n → ∞. In particular, if (ω n ) ∞ n=0 is any sequence with ω n ∈ V n (s) for all sufficiently large n, then Λ(s) coincides with the set Λ ω of accumulation points of the sequence (ω n ).
If F k ∈ F for all k ≥ 0, then additionally:

s). (In particular, A n is closed and
A n = J s \ j≥n V j (s).) In fact, if z belongs to this set, then |f m (z)| ≥ ρ m−n for m ≥ n, where the sequence (ρ ) ∞ =0 depends only on F.
Proof. By Corollary 6.3 the spherical diameter of V n (s) tends to 0 as n → ∞, uniformly in s. In particular, the set Λ ω in (1) is independent of the choice of the sequence (ω n ). Also recall that ζ ∈ V by assumption; if we choose ω n = ζ n , then Λ(s) = Λ(s, ζ) = Λ ω . This proves (1). Part (2) is trivial if F n / ∈ F. Indeed, by assumption on F, we then have F n ⊂ W 0 , τ n+1 (s) = ∞ τ n+1 (s) and τ n+1 (s) ⊂ τ n (s) by definition. So suppose that F n ∈ F. Then F n ⊂ ∞ F n ∪ A Fn and F n ⊂ W 0 ∪ A Fn , where A Fn ⊂ V is the connected set from Lemma 6.9 (iii).
Observe that τ n+1 (s) is the connected component of f −n (F n ) containing ζ n+1 (s), that τ n (s) is the connected component of f −n (W 0 ) containing ζ n (s), and that V n (s) is the connected component of f −n (V ) containing ζ n (s). Let A 1 be the connected component of f −n (A Fn ) contained in V n (s) and let A 2 be the connected component of f −n (A Fn ) contained in τ n+1 (s). Since τ n+1 (s) ⊂ ∞ τ n+1 (s) ∪ A 2 and τ n+1 (s) ⊂ τ n (s) ∪ A 2 , we should show that A 1 = A 2 . Let x ∈ A F ∩ ∞ F n be a point that can be connected to ζ in W 0 ∩ V . Such a point exists by Lemma 6.9 (iv). Let x n be the unique point of f −n (x) in A 1 . Then ζ n (s) and x n belong to the same connected component of f −n (W 0 ); i.e., x n ∈ τ n (s). Now and hence x n ∈ A 2 . We have shown A 1 ∩ A 2 = ∅, and therefore A 1 = A 2 . Also ζ n+1 (s) ∈ A 2 ⊂ V . We have proved both (2) and (3). Now assume that all fundamental domains F n occurring in s = F 0 . . . F n−1 F n . . . belong to F. Let n ≥ 0.
We next prove (4). For n ≥ 0, let X n ⊂ µ σ n (s) ⊂ G σ n (s) be a closed unbounded connected set as in Theorem 2.5 (a). By Theorem 2.5 (c), if R is large enough (depending only on F), then X n can be chosen to contain a point of radius at most R. In particular, X n intersects the set A Fn from Lemma 6.9. So ifX n ⊂ µ s is the connected component of f −n (X n ) contained in τ n+1 (s), thenX n ∩ V n (s) = ∅.
To prove (5), suppose that z ∈ A n . . = G s \ j≥n V j (s). Since z ∈ G s , there is m 0 such that z ∈ ∞ τ m (s) for m > m 0 . If m 0 is minimal with this property, then by (2), By assumption on z, we must have m 0 < n. So, for m ≥ n, we have f m (z) ∈ ∞ F m . Furthermore, by the proof of (2), f m (z) / ∈ A Fm , and thus |f m (z)| > R. So, if R is chosen sufficiently large, f n (A n ) belongs to the set from Theorem 2.5 (d), on which the iterates tend to infinity uniformly and which depends only on F. The same holds for f n (A n ); in particular, this set is contained in G σ m (s) .
Proof of Proposition 6.5. Let F be a finite collection of fundamental domains containing all fundamental domains occurring in s, and also all fundamental domains whose closure intersects D. Let ζ, V and V n be as in Lemma 6.10. Here we assume that R is chosen at least as large as the numbers from Theorem 2.5 (c) and (d).
Throughout the proof we will frequently refer to properties (1)-(5) of filaments G s with s ∈ F ∞ , as established in Lemma 6.10.
By (1), Each of the sets in the intersection on the right is compact and connected by (3); claim (a) follows. We next prove (e). First let U be a neighbourhood of some point z 0 ∈ Λ(s). By (1), there are infinitely many n such that V n ⊂ U . By definition, all points in V n map to the bounded set V after n iterations, and V n contains a point of µ s by (4). Hence f n does not tend to infinity uniformly on U ∩ µ s ⊂ U ∩ G s , as claimed. Observe that this argument also shows that On the other hand, let K ⊂Ĝ s be compact with K ∩ Λ(s) = ∅. Then there is N ≥ 1 such that So f n tends to infinity uniformly on K by (5), and in particular K ⊂ J s . This proves (f). Now we establish the remaining claims in Proposition 6.5. Let z ∈ G s \ Λ(s). By (f), applied to K = {z}, we see that z ∈ G s . Thuŝ Together with (6.1), this proves (b).
Next suppose that z 0 ∈ J s \ G s ; that is, z 0 has address s but is not escaping. There is n 0 such that f n (z 0 ) ∈ J 0 σ n (z) for n ≥ n 0 . Hence |f n (z 0 )| < R by Theorem 2.5 (d). So f n (z 0 ) ∈ V for n ≥ n 0 , and thus z 0 ∈ V n . By (1), this proves Λ(s) = {z 0 }, as claimed.
It remains to prove (d). Recall from Lemma 4.14 that, if s is of disjoint type, then J σ n (s) = G σ n (s) ⊂ F n for all n ≥ 0. Let X be the set from 2.5 (d). Then there is n 0 such that |f n (z)| > R for all n ≥ n 0 and all z ∈ X. Set R . . = max{|f j (z)| : |z| ≤ R and j ≤ n 0 }. By Theorem 2.5 (c), for all n ≥ 0 there is ζ n ∈ J s such that |f n (ζ n )| ≤ R. We claim that |f j (ζ n )| ≤R for all n ≥ n 0 and all j ≤ n. Indeed, let j be minimal such that |f j (ζ n )| > R (if no such j exists, there is nothing to prove). Then f j (ζ n ) ∈ X, and hence we must have j > n − n 0 , and the claim follows by the definition ofR.
Let z 0 ∈ J s be a limit point of the sequence (ζ n ); then all points on the orbit of z 0 have modulus at mostR. The claim now follows from (c).
Remark 6.11 (Coding trees). Fix F, ζ and V as in Lemmas 6.9 and 6.10. For each F ∈ F, we can choose an arc γ F connecting ζ to the point ζ F , and we may assume that these arcs are disjoint except at ζ. For each ζ F 0 and each arc γ F 1 , there is a component of f −1 (γ F 1 ) connecting ζ F 0 to some point ζ F 0 F 1 of f −2 (ζ). By Lemma 6.10, this is precisely the point contained in the fundamental tail at address F 0 F 1 .
Continuing inductively, we obtain an infinite tree with root ζ, whose vertices of depth n > 0 are the elements of f −(n) (ζ) contained in fundamental domains of level n−1 whose addresses contain only entries from F, and whose edges are all components of f −n (γ F ) for some F ∈ F. Recall that the spherical length of these edges tends to zero as n → ∞.
This tree can be considered to be an analogue of the geometric coding tree used by Przytycki [Prz94] in the case of rational functions. We see that, for each address s whose entries are drawn from F, the accumulation set of the filament G s coincides precisely with the accumulation set of a branch of this coding tree. However, we will not use this language in the following.

Separation properties of filaments
We now prove that a filament that lands at a non-escaping point z 0 ∈ C does not separate the plane. This fact is not used in our paper (except to deduce the corresponding parts of Theorems 1.1 and 1.5), but is important for applications.
Theorem 7.1 (Filaments do not separate). Let f be a postsingularly bounded function f , and let s be a bounded external address of f . Assume that G s lands at a point z 0 ∈ C \ G s . Then G s does not separate the plane.
It is plausible that this can be directly deduced from our results and techniques in Section 6; indeed Pfrang does this for postsingularly finite f [Pfr19]. Instead, we deduce Theorem 7.1 by relating landing filaments to Julia continua of disjoint-type entire functions. Recall from Remark 4.15 that a function g is of disjoint type if g is hyperbolic with connected Fatou set, and that a Julia continuum of g is a set of the formĈ = C ∪ {∞}, where C is a connected component of J(g).
Theorem 7.2 (Filaments and Julia continua). Let f be a postsingularly bounded function, and let s be a bounded (resp. periodic) external address of f such that G s lands. If λ is sufficiently small to ensure that g : z → λf (z) is of disjoint type, thenĜ s is homeomorphic to a Julia continuumĈ of g at a bounded (resp. periodic) external address.
The homeomorphism can be chosen to fix ∞, and send z 0 to the unique point of bounded orbit inĈ.
Much is known about the topology of Julia continua of disjoint-type entire functions; see [Rem16]. Theorem 7.2 allows us to transfer this information to landing filaments. In particular, we can easily deduce Theorem 7.1.
Proof of Theorem 7.1, using Theorem 7.2. Let g be a disjoint-type function as in Theorem 7.2. Then the Fatou set F (f ) is connected and non-empty by definition. Hence J(f ) is a nowhere dense set that does not separate the plane, so no subset of J(f ) separates the plane.
So by Theorem 7.2, the setĜ s is homeomorphic to a non-separating plane continuum. It is well-known that being a one-dimensional non-separating plane continuum is a topological property. (Indeed, a one-dimensional plane continuum is non-separating if and only if it is tree-like; see [Bin51, Theorem 6] and [Mań12, Theorem 1.5]. Compare also [JT02].) SoĜ s is also non-separating.
Proof of Theorem 7.2. By [BK07, Example on p. 392], for λ ∈ C small enough the function g is indeed of disjoint type; we fix such λ in the following. By [Rem09, Theorem 1.1], there is a map ϑ defined on which is a conjugacy between f on J ≥R (f ) and g on ϑ(J ≥R (f )). Furthermore, ϑ extends continuously to ∞ with ϑ(∞) = ∞ -in particular, ϑ maps escaping points of f to escaping points of g -and J ≥Q (g) ⊂ ϑ(J ≥R (f )) for some Q [Rem09, Lemma 3.3].
Assuming that R > 0 is sufficiently large, the proof of [Rem09, Theorem 3.2] furthermore implies the following. For any external address address s of f , there is an external addresss of g such that . If s is bounded (resp. periodic), then so iss. We can extend ϑ| J ≥R (f )∩I(f ) to a bijection ϑ : I(f ) ∪ {∞} → I(g) ∪ {∞} by defining ϑ(z) . . = g −ñ s (ϑ(f n (x))). The value ϑ(z) is independent of n and, in particular, agrees with the original value when z ∈ J ≥R (f ) ∪ I(f ). This follows from (7.1) and Proposition 4.4 and the conjugacy relation for ϑ. Note that we do not claim that this bijection is continuous on I(f ). Now suppose that s is bounded and the filament G s (f ) lands at a point x 0 ∈ C \ G s . Note that the filament Gs of g also lands at some point y 0 ∈ C of bounded orbit since the function g is of disjoint type. (Recall Proposition 6.5 (d).) Consider the closuresĜ s (f ),Ĝs(g) of G s (f ) and Gs(g) inĈ. For n ∈ N define By definition, X n ⊃ X n−1 , and ϑ is continuous when restricted to X n . We claim that ϑ is continuous on X . . = n X n = G s (f ) ∪ {∞}.
Let x ∈ X. By assumption, x / ∈ Λ(s) = {x 0 }. Hence by Corollary 6.6, z has a neighborhood U in X on which the iterates escape to infinity uniformly. Then, for sufficiently large n, f n (U ) ⊂ J 0 s (f ), and hence U ⊂ X n . Since ϑ is continuous on X n and U is a neighbourhood of x, ϑ is continuous at x.
Moreover, ϑ −1 is continuous on the sets Y n . . = g −n s (J 0 σ n (s) (g) ∩ J ≥Q (g)) ∪ {∞}. Hence, by the same argument as for f , the map ϑ −1 is continuous on Y = Gs(g) ∪ {∞}, and ϑ : X → Y is a homeomorphism. In particular, it extends to a homeomorphism between their respective one-point compactificationsĜ Remark. It is plausible that the homeomorphism in Theorem 7.2 is ambient; i.e., it extends to a homeomorphism of C onto itself.

Landing theorems for filaments
With the definition of filaments in Definition 4.2 and their accumulation sets in Definition 6.4 we can now state the main result of our paper. Recall that a filament G s is periodic if the address s is periodic under the shift map. Equivalently, G s is periodic under the action of f as a subset of C.
Theorem 8.1 (Douady-Hubbard landing theorem for filaments). Let f be a transcendental entire function whose post-singular set P(f ) is bounded.
Then every periodic filament of f lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic periodic point of f is the landing point of at least one and at most finitely many periodic filaments.
We also obtain a corresponding theorem about landing properties at more general hyperbolic sets.
Theorem 8.2 (Landing at hyperbolic sets). Let f be a transcendental entire function with bounded postsingular set. Moreover, suppose that K is a hyperbolic set for f . Then every point z ∈ K is the landing point of a filament at some bounded external address s; if z is periodic, then so is s.
In fact, we establish the following more precise version of Theorem 8.2. Note that the space of external addresses is equipped with the product topology; this is the same as the order topology arising from the cyclic order on addresses.
Theorem 8.3 (Accessibility of points in hyperbolic sets). Let K be a hyperbolic set of a postsingularly bounded entire function f . Then there exists a compact and forwardinvariant set S K of bounded addresses of f , with the following properties.
(a) For every s ∈ S K , the filament at address s lands at a point z 0 (s) ∈ K.
(b) The function S K → K; s → z 0 (s) is surjective and continuous. In particular, every point of K is the landing point of a filament. (c) If z 0 is periodic, then all bounded-address filaments landing at z 0 are periodic with the same period, and the number of such filaments is finite. (d) The filaments at addresses in S K land uniformly at K, in the following sense.
Let ζ ∈ W 0 and let ε > 0. Then there is n 0 such that dist(ζ n (s), K) ≤ ε for all n ≥ n 0 and all s ∈ S K . (Recall from Definition 6.1 that ζ n (s) = f −n s (ζ).) Note that we do not claim that S K can be chosen to consist of all addresses of filaments landing at points of K. In particular, we do not prove that the function s → z 0 (s) is continuous on the latter set.
The remainder of the paper will be dedicated to the proofs of these theorems. Let us first show that they imply the theorems stated in the introduction.
Proof of Theorem 1.1, using Theorem 8.1. Let ζ be a repelling or parabolic periodic point. By Theorem 8.1, ζ is the landing point of a periodic filament G s . Let us set A . . = G s , and let p be the period of s. Recall that G s ⊂ I(f ) by definition, and is unbounded and connected by Proposition 4.10. Since G s lands at ζ, we see from Corollary 6.6 that A = A ∪ {ζ} and that, for any neighborhood U of ζ, f n → ∞ uniformly on A \ U . Furthermore, by Theorem 7.1, A does not separate the plane. This proves that A satisfies properties (a) and (c) of Theorem 1.1. Since p is the period of s, we have Proof of Theorem 1.4, using Theorem 8.1. Recall that every periodic hair is a periodic filament (Proposition 5.5), and for criniferous functions, every periodic filament is a periodic hair (Corollary 5.4). Moreover, by Corollary 6.7 such a filament lands if and only if the corresponding hair lands in the sense of Definition 1.3.
Thus Theorem 1.4 follows immediately from Theorem 8.1.
Proof of Theorem 1.5, using Theorem 8.3. Define A . . = {G s : s ∈ S K }, where S K is as in Theorem 8.3. Properties (a), (c) and (e) in Theorem 1.5 follow immediately, as in the proof of Theorem 1.1. Furthermore, if f is criniferous, then every element of A is an arc connecting its landing point ζ(A) to ∞, by Corollary 6.7. Thus it remains to establish (d). Observe that, since S K is compact and forwardinvariant, only finitely many different fundamental domains can occur within the addresses in S K . Let F be the collection of all these fundamental domains, as well as of all fundamental domains whose closure intersects D. Let V be as in Lemma 6.10, and let n 0 be as in part (d) of Theorem 8.3 for ε/2. By Lemma 6.10, we can find n 1 ≥ n 0 such that diam V n (s) < ε/2 for n ≥ n 1 . Thus dist(z, K) < ε for all z ∈ V n (s).
Let z ∈ G s for some s ∈ S K , and suppose that dist(z, K) ≥ ε. Then, by the above z ∈ G s \ j≥n 1 V j (s). By (5) of Lemma 6.10, the set of points with this property escape to infinity at a rate that depends only on F and on n 0 (and hence only on ε). This completes the proof.
Remark 8.4 (Shortcut to the landing theorems). Observe that Definition 6.1, concerning accumulation sets, requires only the definition of the fundamental tails τ n (s) associated to an address s, rather than any properties of the filament G s itself. Moreover, a key point in our proofs of Theorems 8.1, 8.2 and 8.3 is that we can ignore the fine structure of filaments and use only the sets V n from Lemma 6.10. We only require properties (1), (2) and (3) of this lemma, all of which are likewise independent of the construction and analysis of filaments in Sections 2, 4 and 5.
Hence it would be possible to prove these theorems without requiring any results from those sections, and using only the elementary parts of Section 6. Furthermore, while we used properties of filaments above to deduce the statements of Theorems 1.1 and 1.5, many of these can be established more easily a posteriori for the filaments under consideration, using the additional information that these filaments land. Nonetheless, the general material concerning filaments is crucial to the interpretation of our results as a natural extension of the classical Douady-Hubbard landing theorem for functions without hairs.

Periodic filaments land
We now prove the first half of Theorem 8.1, which we restate here as follows.
Theorem 9.1 (Landing of periodic filaments). Let f : C → C be a transcendental entire function with bounded postsingular set.
Then every periodic filament of f lands at a repelling or parabolic periodic point of f , where the period of the landing point divides that of the filament.
Proof of Theorem 9.1. Let G s be a periodic filament. Recall from Observation 4.13 that any filament of f is also a filament of any iterate of f (and vice versa). Hence, it is no loss of generality to assume that s is fixed by σ; i.e., s = F F F F . . . , where F is a fundamental domain of f .
Fix ζ and V as in Lemmas 6.9 and 6.10. Let τ n be the fundamental tail of level n associated to s, and let ζ n be the unique element of f −n (ζ) in τ n .
By Lemma 6.10 (1), the spherical diameter of γ n tends to zero as n → ∞. Therefore the set Λ(γ) of accumulation points of γ(t) as t → −∞ is precisely the accumulation set Λ(G s ) = Λ(s, ζ) of G s in the sense of Definition 6.1.
Claim. As t → −∞, γ(t) converges to a fixed point of f .

Proof.
Recall that the spherical distance between ζ n and ζ n+1 tends to zero as n → ∞, and f (ζ n+1 ) = ζ n . Hence, by continuity, any finite point of Λ(γ) = Λ(s, ζ) is a fixed point of f . Since the set of fixed points is discrete in C, and Λ(γ) is connected, the latter set is a singleton, whose sole element is either a fixed point or ∞. We must exclude the second possibility.
So suppose, by contradiction, that γ(t) → ∞ as t → −∞. Recall that Ω is the unbounded connected component of C \ P(f ). Let ρ n . . = Ω (γ n ) be the hyperbolic length of γ n in Ω. Since γ n → ∞ and the postsingular set is bounded, formula (3.2) of Proposition 3.1 implies that ρ n+1 ≤ ρ n /2 for all sufficiently large n. It follows that the hyperbolic length of γ is bounded. As the hyperbolic metric on Ω is complete, this contradicts our assumption that γ tends to ∞.
The classical snail lemma of Douady and Sullivan (see [DH85, Exposé VIII, Proposition 2, p. 60] or [Mil06,Lemma 16.2]) shows that the limit point of γ is either repelling or parabolic with multiplier 1, and the proof is complete.
Remark. In fact, the above claim follows already from [Rem08, Theorems B.1 and B.2]. Here Theorem B.1 is a hyperbolic expansion argument going back to the proof by Douady and Hubbard [DH85,Exposé VIII] of the first half of the landing theorem for polynomials. On the other hand, the proof of [Rem08,Theorem B.2], which shows that γ(t) cannot converge to infinity as t → −∞, used the notion of "extendability", which was developed for more general purposes in [Rem08]. For the reader's convenience, we gave the complete and much simpler proof above, in the spirit of Deniz [Den14].
It is possible that the landing point in Theorem 9.1 is also the landing point of other filaments. However, as we now observe, there can only be finitely many of these, and they all need to be periodic. The idea of the proof is very similar to the polynomial case [Mil06,Lemma 18.12], but we need to take into account the non-compactness of the space of addresses. Compare also [RS08a, Lemma 3.2] for the case of exponential maps.
Lemma 9.2. Let f be a transcendental entire function with bounded postsingular set, and let z be the landing point of a filament with periodic address. Then the number of bounded-address filaments landing at z is finite, and their addresses are all periodic of the same period.
Proof. Let s 0 be the address of the periodic filament landing at z; we may assume that its period p is minimal with this property. Replacing f by f p , we may then assume that p = 1. Let S z be the set of bounded external addresses s for which G s lands at z.
Claim. There is a finite collection F of fundamental domains with the following property. Every address s ∈ S z contains some element of F infinitely many times.
Proof. Let F consist of all fundamental domains F such that either D ∩F = ∅, or z ∈ F . Clearly F is finite. Suppose that s = F 0 F 1 F 2 . . . is such that F n / ∈ F for n ≥ n 0 . Then σ n 0 (s) is of disjoint type, and z / ∈ F n 0 ⊃ f n 0 (G s ). In particular, s does not land at z. So any address in S z contains infinitely many entries from F; since the latter set is finite, at least one of these is itself repeated infinitely many times.
Since f maps a neighborhood of z to another neighborhood of z as an orientationpreserving homeomorphism, it preserves the cyclic order of the filaments landing at z. As remarked at the end of Section 13, this implies that f also preserves the cyclic order of these filaments at ∞. In other words, the shift map σ : S z → S z is injective and preserves the cyclic order of addresses on S z .
Recall that s 0 is a fixed address, say s 0 = F 0 F 0 F 0 . . . . Hence σ also preserves the (linear) order <, where s <s means that s 0 ≺ s ≺s in the cyclic ordering. So if s = F 0 F 1 F 2 . . . is an element of S z , then (σ n (s)) ∞ n=0 is monotone. If F ∈ F is the domain from the claim above, then we clearly must have F n = F for all sufficiently large n. By injectivity of σ, we conclude that s = F F F . . . is itself a fixed address. As F is finite, the proof is complete.
Remark 9.3. Recall that we defined filaments only for maps with bounded postsingular sets, and landing of filaments only for filaments at bounded addresses. This allows us to state the lemma in the above simple form, which is all that will be required for the purpose of our main results.
However, observe that the proof of the lemma is purely combinatorial, and does not utilise either assumption in an essential manner. In particular, let f ∈ B be arbitrary (not necessarily with bounded postsingular set), and let z 0 ∈ J(f ) be a fixed point of f . Suppose that S z is a forward-invariant set of external addresses s of f , and that for every s ∈ S there is an unbounded connected set A s with the following properties: • the cyclic order of the sets A s at infinity agrees with the cyclic order of their external addresses; • z 0 ∈ A s for all s ∈ S z ; • A s ∩ As = ∅ for s =s; • A s does not separate the plane; • f (A s ) = A σ(s) . If S contains a periodic element, then it follows as above that S z is finite and contains only periodic addresses. In particular, for any f ∈ B, the landing point of a periodic hair cannot be the landing point of a non-periodic hair.

Landing at hyperbolic sets
Proof of Theorem 8.3. Let K be a hyperbolic set of f . Replacing f by a sufficiently high iterate, there is a neighbourhood U of K such that |f (z)| ≥ 2 for all z ∈ U . (It is easy to see that proving the theorem for an iterate of f also establishes it for f itself; we leave the details to the reader.) We may additionally assume that the disc D in the definition of fundamental domains is chosen so large that K ⊂ D. Finally, by Corollary 6.3, it is enough to prove (d) for some specific choice of ζ ∈ W 0 . Therefore we may fix ζ belonging to an unbounded connected component of W 0 ∩ f −1 (D), as required in the hypothesis of Lemma 6.9.
Set δ . . = dist(K, ∂U ). For z ∈ K, define B 0 (z) . . = B(z, δ), and let B n (z) denote the connected component of f −n (B 0 (f n (z))) containing z. Then B n+1 (z) ⊂ B n (z) for all n, and f : B n+1 (z) → B n (f (z)) is a conformal isomorphism. For n ≥ 0, define Then U n+1 ⊂ U n ⊂ U and f (U n+1 ) = U n for all n. Clearly K = n≥0 U n = n≥0 U n . By the blowing-up property of the Julia set (see e.g. [Bak84, Lemma 2.2]), and compactness of K, there is some N 1 with the following property: if n ≥ N 1 and z ∈ K, then f −n (ζ) ∩ B 0 (z) = ∅. In particular, for all such n and z there is a finite external address of length n such that ζ n (s) ∈ B 0 (z), and hence τ n (s) ∩ B 0 (z) = ∅. Define Now let V be as in Lemma 6.9 for the finite collection F of fundamental domains that intersect D. Similarly as in Lemma 6.10, if s is an infinite external address or a finite external address of length at least n, we define V n (s) to be the unique component of f −n (V ) containing ζ n (s) . . = f −n s (ζ) ∈ τ n (s). By Lemma 6.2, there exists N ≥ N 1 with the following property. Suppose that n ≥ N and that s is a finite or infinite external address of length at least n with V n (s) ∩ U 0 = ∅.
For n ≥ N and z ∈ K, we now define S n (z) to be the set of finite external addresses s of length n for which the tail τ n (s) intersects B n−N (z). Observe that is finite for every n by Lemma 2.1. We also define S K to be the set of infinite addresses s such that every prefix of length n ≥ N of s is an element of S n . In order to show that this set has the properties asserted in Theorem 8.3, we investigate the sets S n (z) more closely.
Claim 1. The following hold for all n ≥ N and all z ∈ K.
Nb. In (ii), we would like to claim that σ : S n+1 (z) → S n (f (z)) is a bijection. However, it is conceivable that a tail τ of level n intersects B n−N (f (z)) in more than one connected component, and that τ ∪ B n−N (f (z)) surrounds a singular value of f . In this situation, there may be two different components of f −1 (τ ) that intersect B n+1−N (z), and σ may therefore not be injective on S n+1 (z). Nonetheless, Claim 3 below implies that this situation can arise only for small n. Hence, for sufficiently large n, the map σ : S n+1 (z) → S n (f (z)) will turn out to be a bijection after all (see Claim 4).
Claim 2. There is a finite collectionF ⊃ F of fundamental domains such that all entries of addresses in n≥N S n are inF. Now consider the directed graph G whose vertices are the elements of and which contains an edge from π n (s) to s for every s ∈ S n+1 . Note that G is a locally finite, infinite graph on countably many vertices. For z ∈ K, let G z be the induced subgraph of G whose vertices are the elements of n≥N S n (z). By (iii) and (iv) of Claim 1, we can apply König's lemma, and G z contains an infinite path for every z ∈ K.
Recall that S K is the set of infinite external addresses s such that π n (s) ∈ S n for all n ≥ N . If s ∈ S K , the sequence (π n (s)) ∞ n=N forms an infinite path in G. Conversely, every infinite path in G determines an associated address s ∈ S K . For z ∈ K, denote by S z the set of all s ∈ S K with π n (z) ∈ S n (z) for all n ≥ N . By the above, S z = ∅ for all z ∈ K.
The set S K is shift-invariant by part (ii) of Claim 1. Furthermore, S K is contained in the compact set of addresses all of whose entries are taken fromF; we need to show that S K is itself compact. Suppose that (s k ) ∞ k=0 is a sequence of addresses in S K converging to some address s. Then the prefixes π n+1 (s) and π n+1 (s k ) agree for all sufficiently large k, and in particular π n (s) and π n+1 (s) are two vertices of G connected by an edge. It follows that s is indeed represented by an infinite path in G, and hence s ∈ S K as required.
To prove claim (a) of Theorem 8.3, we must show that G s lands at a point z 0 (s) ∈ K for all s ∈ S K . By Claim 3, there is some n 0 such that, for all n ≥ n 0 and all s ∈ S K , there is z ∈ K such thatṼ n (s) ⊂ B 0 (z). In particular, diamṼ n 0 (s) ≤ 2δ, and by expansion of f on U , we conclude that In particular, ζ n (s) is a Cauchy sequence, and hence convergent. So G s lands at a point z 0 (s) with (10.4) dist(z 0 (s),Ṽ n (s)) ≤ 2 n 0 −n+1 · δ, for all n ≥ n 0 . It is clear from Claim 3 that the landing point z 0 (s) belongs to K. Furthermore, if z ∈ K and s ∈ S z , then z 0 (s) = z by Claim 3. As S z = ∅ for all z ∈ K, this shows that the function s → z 0 (s) is surjective. To prove continuity, suppose that s,s ∈ S K agree in the first n ≥ n 0 entries. ThenṼ n (s) =Ṽ n (s), and hence dist(z 0 (s), z 0 (s)) ≤ 3 · 2 n 0 −n+1 · δ by (10.3) and (10.4). This completes the proof of (b).
Since s is periodic under ϕ, of period k 0 , we conclude that s k = π (k+1)n 0 (s k+k 0 ). Hence there is an infinite path in G z passing through the vertices s j·k 0 , j ≥ 0. The associated address is the periodic sequence (ϕ k 0 −1 (s) . . . ϕ 1 (s)s) ∞ , and the proof is complete.
We note the following corollary, which proves the accessibility of certain singular values. For definitions, we refer to [RvS11].
Corollary 10.1 (Accessibility of non-recurrent singular values). Let f be a postsingularly bounded transcendental entire function, and let v ∈ J(f ) be a non-recurrent singular value for f whose forward orbit does not pass through any critical points. Suppose that the ω-limit set of v does not contain parabolic points, and does not intersect the ω-limit set of a recurrent critical point or of a singular value contained in a wandering domain. Then there is a bounded-address filament of f that lands at v.
Proof. By [RvS11, Theorem 1.2], if the postsingular set is bounded then any forward invariant compact subset of the Julia set is hyperbolic provided it does not contain parabolic points, critical points, or it intersects the ω-limit set of a critical point or of a singular value contained in wandering domains. Hence P (a) := n f n (a) is hyperbolic and every point in P (a) is the landing point of a filament.

Landing at parabolic points
We now complete the proof of our analogue of the Douady-Hubbard landing theorem, Theorem 8.1, by showing that parabolic periodic points are also accessible by filaments.
Theorem 11.1 (Parabolic points are accessible by filaments). Let f ∈ B with bounded postsingular set, and let z 0 be a parabolic periodic point. Then there is a periodic filament of f that lands at z 0 .
Let f be as in the statement of the theorem. By passing to an iterate, we may assume that f (z 0 ) = 1. So z 0 is a multiple fixed point of f , say of multiplicity m + 1 for f . Then there are m unit vectors v 1 . . . v n , called repelling directions at z 0 . Any backward orbit of f converging to z 0 must asymptotically converge to z 0 along one of these directions; see [Mil06,Lemma 10.1]. Similarly, there are n attracting directions w n such that any forward orbit (f n (z)) ∞ n=0 converging to z 0 must converge to z 0 along one of these attracting directions w n .
Let U be a small simply connected neighborhood of z 0 on which f is univalent, and let ψ : f (U ) → U be the branch of f −1 that fixes z 0 . A petal for an attracting (resp. repelling) direction w (resp. v) is an open set P ⊂ U containing z 0 on its boundary, such that (1) f (P ) ⊂ P (resp. ψ(P ) ⊂ P ); (2) an orbit z → f (z) → . . . (resp. z → ψ(z) → . . .) is eventually absorbed by P if and only if it converges to z 0 from the direction w (resp. v). Petals for a given repelling or attracting direction are far from unique. For each repelling direction v, we can choose a repelling petal P v for v which is simply connected, and such that ψ(P v ) ⊂ P v ∪ {z 0 } and ψ n | Pv → z 0 uniformly on P v . Similarly, for each attracting direction w we choose a simply connected attracting petal P w such that f n → z 0 uniformly on P w . We furthermore require that the union of these n attracting and n repelling petals forms a punctured neighborhood of z 0 (see Definition 10.6 and Theorem 10.7 in [Mil06] and the subsequent discussion).
Definition 11.2 (Landing of filaments along a repelling direction). Let ζ ∈ W 0 , let G s be a periodic filament of f , and let v be a repelling direction at z 0 . We say that G s lands at z 0 along v if the backwards orbit (ζ n (s)) ∞ n=1 converges to z 0 along the direction v. We remark that it is not difficult to see that this is equivalent to requiring that V n (s) ⊂ P v for all sufficiently large n, where V is as in Lemma 6.10. In particular, the definition is independent of the choice of the base point ζ.
The following establishes Theorem 11.1.
Proposition 11.3 (Accessibility along repelling directions). Let v be a repelling direction of f at z 0 . Then there is at least one periodic filament landing at z 0 along v.
Proof. Let ζ ∈ W 0 and let V be as in Lemma 6.10, with F once again the finite collection of fundamental domains whose closure intersects D. Since V ⊂ C\P(f ) and z 0 ∈ P(f ) ⊂ D, we may assume that the repelling petals P v and attracting petals P w chosen above all have closures disjoint from V ∪ W 0 . Let us define B i . . = ψ i (P v ) for i ≥ 0. Let A be the union of the attracting petals P w . Since the union of attracting and repelling petals is a punctured neighbourhood of z 0 , all points of ∂B 0 that are sufficiently close to z 0 must lie in A ∪ {z 0 }. So ∂B 0 \ (A ∪ {z 0 }) is a compact set disjoint from B 1 , and Since B 1 intersects J(f ), there is an N 1 such that f −n (ζ) ∩ B 1 = ∅ for n ≥ N 1 . In particular, there exists some finite external address of length n such that τ n (s) ∩ B 1 = ∅. By Lemma 6.2, there is N ≥ N 1 such that, for all n ≥ N and all infinite external addresses s with V n (s) ∩ B 0 = ∅, diam V n (s) < ε whenever n ≥ N . Observe that V n (s) ∩ A = ∅ by our choice of petals. In particular, if n ≥ N and V n (s) ∩ B 1 = ∅, then V n (s) ∩ ∂B 0 = ∅, and hence V n (s) ⊂ B 0 .
As in the proof of Theorem 8.3, for n ≥ N we define S n to consist of those finite external addresses of length n for which τ n (s) intersects B n−N . The remainder of the proof then proceeds analogously.
In the case that all periodic filaments are hairs (for example, if f is criniferous), our Proposition 11.3 is a corollary of the Main Theorem in [BF15] (since the hypothesis that periodic rays land is implied by assuming bounded postsingular set), with a completely different proof. We remark that it is plausible that the results of [BF15] can also be extended to non-criniferous functions, using filaments instead of hairs.
Proof of Theorem 8.1. That every periodic filament lands at a repelling or parabolic point was proved in Theorem 9.1. Let z 0 be a repelling or parabolic point. If z 0 is repelling, then the orbit of z 0 is a hyperbolic set, and it follows from Theorem 8.3 that z 0 is the landing point of a periodic filament. If z 0 is a parabolic point, then this fact follows from Theorem 11.1. By Lemma 9.2 there are only finitely many filaments landing at z 0 and they are all periodic of the same period.

Filaments landing together at points in a hyperbolic set
Recall from Theorem 8.1 that, for a repelling periodic point z 0 of a postsingularly bounded function f , the number of filaments landing at z 0 is finite. In the polynomial case, this holds also for every point z 0 in a hyperbolic set K of f . It is plausible that this remains true also in the transcendental entire case. For postsingularly bounded exponential maps, the claim is proved in [BL14,Proposition 4.5], where it is proved that the number of hairs in question is even uniformly bounded (depending on K). However, the proof uses the fact that postsingularly bounded exponential maps are non-recurrent, and hence the postsingular set is itself a hyperbolic set.
Here we shall be content with proving that the number of filaments of a postsingularly bounded function f landing at a given point of a hyperbolic set is (pointwise) finite, in the important special case where f belongs to the Speiser class; i.e., the set of singular values S(f ) is finite.
Theorem 12.1 (Finitely many filaments landing together). Let f be a postsingularly bounded entire function with finitely many singular values. Suppose that z 0 ∈ J(f ) \ I(f ) is neither a Cremer periodic point nor a preimage of such. Then the number of boundedaddress filaments G s landing at z 0 is finite.
The assumption that f is postsingularly bounded implies, via Theorem 8.1 and Lemma 9.2, that one can restrict to the case where z 0 is not (pre-)periodic. In addition, this hypothesis and the restriction to bounded addresses s ensure that we can speak about the filaments G s and their landing properties at all. (Recall Remark 9.3.) However, the argument can be applied also in more general circumstances. For example, the same proof can be used to show the following: if S(f ) is finite (but the postsingular set is not necessarily bounded), and z 0 ∈ J(f ) \ I(f ) is not periodic and also is the landing point of at least one bounded-address hair, then the number of hairs landing at z 0 is finite, and all of them have bounded addresses.
Remark 2. The assumption that z 0 / ∈ I(f ) is made to avoid complications in the case where z 0 itself belongs to one of the filaments landing at z 0 . An escaping point in a bounded-address filament cannot in fact be accessible by the same or another filament, due to the presence of other filaments accumulating on it from both sides; recall the proof of Corollary 6.7, and compare [Rem16, Theorem 2.3]. Assuming this fact, the assumption that z 0 / ∈ I(f ) could be omitted.
Corollary 12.2 (Finiteness of filament portraits at hyperbolic sets). Let f be a postsingularly bounded entire function with finitely many singular values. If K is a hyperbolic set for f , then every point z 0 ∈ K is the landing point of at least one and at most finitely many bounded-address filaments.
Proof. By definition, a hyperbolic set contains no Cremer periodic points, their preimages, or escaping points. Hence this is a combination of Theorems 8.2 and 12.1.
We now fix a postsingularly bounded entire function f with #S(f ) < ∞ for the remainder of the section. The key property that we need to establish in the proof of Theorem 12.1 is that the addresses of filaments landing at z 0 are uniformly bounded, in the sense that they all take their entries from a common finite family of fundamental domains. This is the content of the following lemma.
Lemma 12.3. Let F 1 be a finite collection of fundamental domains for f . Then there exists another finite collection F 2 ⊃ F 1 of fundamental domains such that the following holds. Suppose that s 1 takes only entries from F 1 and that G s 1 lands at a non-escaping point z 0 ∈ C. If s 2 is bounded and G s 2 also lands at z 0 , then all entries of s 2 belong to F 2 .
Let us suppose for a moment that the function f is criniferous. Then the idea of the proof of Lemma 12.3 can be described as follows. If G s 1 and G s 2 land together at a point z 0 , the filaments G σ(s 1 ) and G σ(s 2 ) also land together at f (z 0 ), by continuity of f . There is a branch ϕ of the inverse of f on the hair G σ(s 1 ) that maps it to G s 1 . The curve G s 1 ∪ {z 0 } ∪ G s 2 is then obtained by analytic continuation of ϕ along the image curve G σ(s 1 ) ∪ {f (z 0 )} ∪ G σ(s 2 ) . For this reason, the homotopy class of the latter curve in C \ S(f ), together with the first entry of s 1 , essentially determines the first entry of s 2 . As different pairs of hairs landing at the same point are disjoint, and S(f ) is finite, there are only finitely many possible such homotopy classes. The claim follows. In order to make this argument precise in the general case, i.e. where the filaments are not necessarily hairs, we should clarify what we mean by "homotopy classes". Let us fix the postsingularly bounded function f with finite singular set for the remainder of the section.
Let Γ be the class of continuous curves γ : R → C \ S(f ) that tend to infinity within W 0 in both directions. We shall say that such curves γ 1 and γ 2 are homotopic (in Γ) if they are homotopic (relative to their endpoints at infinity) in C \ (S(f ) ∪δ), for some infinite pieceδ of the curve δ used in the definition of fundamental domains.
Similarly, letΓ denote the set of curves connecting a finite endpoint z 0 ∈ C (possibly belonging to S(f )) to infinity within C \ S(f ), again tending to infinity within W 0 . Then we analogously define homotopy classes for curves inΓ having the same endpoint.
We can now introduce a convenient notion for homotopy classes of bounded-address filaments. Suppose that s is a bounded external address, and that the filament G s lands at a point z 0 ∈ C \ G s . Then there is an infinite pieceδ of δ not intersecting G s . It follows that there is a Jordan curve J, passing through infinity, that separates G s from δ and all of the finitely many points of S(f ) \ {z 0 }. Let γ be an arc connecting z 0 to infinity in the connected component V of C \ J containing z 0 . The homotopy class of G s is the homotopy class of γ inΓ, as defined above.
Note that this homotopy class depends only on s. Indeed, suppose thatṼ is a second domain as above, andγ ⊂Ṽ connects z 0 to infinity. Since G s ⊂ V ∩Ṽ = . . U , this open set U contains a curve α connecting z 0 to infinity. (See e.g. [Rem08, Lemma A.1].) Since V is simply connected, α is homotopic to γ in V , and hence inΓ. For the same reason, α is homotopic toγ.
Observation 12.4 (Disjoint curves representing homotopy classes). Let S 1 , . . . , S n be finitely many different bounded external addresses, such that each G S j lands at a nonescaping point z j ∈ C for all j. (We do not assume that all z j are distinct.) Then there exists a collection (γ j ) n j=1 of arcs to infinity, with γ j in the homotopy class of G s , such that these arcs are pairwise disjoint apart from common endpoints.
Proof. Similarly as above, we can find a finite collection of Jordan curves (J ) m =1 , disjoint fromδ ∪ n j=1 G S j , such that any two distinct landing points z j 1 and z j 2 are separated by some J . (Here, as above,δ is an infinite piece of the curve δ that does not intersect any of the filaments under consideration.) Let V j be the connected component of C \ m =1 J containing z j . We can choose the curve Γ in the definition of the homotopy class of G S j in such a way that Γ additionally separates z j from ∂V j . This shows that the γ j may be chosen disjoint, except possibly for those having a common endpoint. But any curves with a common endpoint belong to the same V j , and therefore can also be moved by homotopy within the simply connected domain V j to be disjoint, except at that endpoint. This completes the proof.
If two bounded-address filaments G s 1 and G s 2 land at a common non-escaping point z 0 , we shall refer to these two filaments as a filament pair. If z 0 / ∈ S(f ), then we can form a curve in Γ by combining two arcs γ 1 and γ 2 , in the homotopy class of G s 1 and G s 2 , respectively. The corresponding homotopy class is called the homotopy class of the filament pair.
Lemma 12.5 (Finitely many homotopy classes). There are only finitely many different homotopy classes of filament pairs not landing at singular values.
Similarly, for any z 0 ∈ C, there are only finitely many homotopy classes of filaments landing at z 0 .
Proof. The curves representing the homotopy class of two different filament pairs are disjoint, except for the endpoints at infinity, and possibly a single additional point (if the filament pairs land at the same point). Also recall that neither curve self-intersects. It follows that, if both curves wind around the same collection of singular values in positive orientation, and both either surround or do not surround an infinite piece of δ, they represent the same homotopy class. As there are only finitely many singular values, the set of homotopy classes is finite.
The second claim follows in the same manner.
The following is immediate from the homotopy lifiting property.
Observation 12.6 (Connecting fundamental domains). Let γ ∈ Γ. Suppose that F is a fundamental domain, and letγ : (−∞, ∞) → C \ f −1 (S(f )) be the unique lift of γ under f such thatγ(−t) ∈ F for all sufficiently large t. Then there is a fundamental domainF such thatγ(t) ∈F for large t, andF depends only on F and the homotopy class of γ in Γ.
Similarly, let γ ∈Γ connect a finite point z 0 ∈ C to ∞. If F is a fundamental domain, andγ is the lift of γ under f that tends to infinity within F , then the finite endpoint w 0 ofγ depends only on F and the homotopy class of γ inΓ.
Proof of Lemma 12.3. LetF 2 consist of all domainsF as in Observation 12.6, where F ranges over the finitely many elements of F 1 , and the homotopy class of γ ranges over the finitely many homotopy classes of filament pairs of f . Now suppose that G s 1 and G s 2 form a filament pair, with F 1 0 ∈ F 1 . Let z 0 be the common landing point of the two filaments. If f (z 0 ) / ∈ S(f ), then it follows from Observation 12.6 (applied to the curve γ σ(s 1 ) ∪ {f (z 0 )} ∪ γ σ(s 2 ) ) that F 2 0 ∈F 2 . On the other hand, suppose that s = f (z 0 ) ∈ S(f ). Then, by Observation 12.6, z 0 depends only on the homotopy class of γ σ(s 1 ) , and the entry F 1 0 . Hence, for each singular value s, there are only finitely many possible preimages z 0 that can arise as landing points of filaments whose first entry is in F 1 .
Consider such z 0 , and the curve γ = γ σ(s 2 ) ∈Γ connecting f (z 0 ) to ∞. Then γ has d different lifts starting at z 0 , where d is the local degree of f at z 0 , tending to infinity within fundamental domainsF 1 , . . . ,F d . This collection of fundamental domains depends only on the homotopy class of γ by Observation 12.6. In particular, there is a collection F(z 0 ) of at most m·d fundamental domains, where m is the (finite) number of homotopy classes of filaments connecting s to ∞, such that F 2 0 ∈ F(z 0 ) whenever s 2 is as above. Recall that there are only finitely many singular values s, and for each of these only finitely many preimages z 0 as above. Thus we can add the finitely many sets F(z 0 ) tõ F 2 to obtain a set F 2 with the desired property.
Proof of Theorem 12.1. If z 0 is (pre-)periodic, then by assumption f n (z 0 ) is a repelling or parabolic periodic point for some n ≥ 0. As remarked above, in this case the conclusion of the theorem holds by Theorem 8.1 and Lemma 9.2. Hence we can assume that z 0 is not a pre-periodic point.
Since f is postsingularly bounded, every orbit of f passes through only finitely many critical points. Indeed, points with unbounded orbits cannot go through critical points at all, and the intersection of any bounded orbit with the (discrete) set of critical points is finite. Hence, passing to a forward iterate, we may additionally assume that the forward orbit of z 0 does not contain a critical point. Let F 1 be the set of fundamental domains occurring in s, and let F 2 be the set whose existence is guaranteed by Lemma 12.3; say F 2 = {F 0 , F 1 , . . . , F m−1 }, where we assume that F 0 ≺ F 1 ≺ · · · ≺ F m−1 ≺ F 1 with respect to the cyclical order at infinity.
Let X be the set of points on the unit circle S 1 = R/Z having an (m+1)-ary expansion that contains only the entries 0, . . . , m − 1. Via the (m + 1)-ary expansion, this set is order-isomorphic to {0, . . . , m − 1} N , which in turn is clearly order-isomorphic to F N 2 . Let ϕ : F N 2 → X be this order-isomorphism; then ϕ conjugates the shift on F N 2 to the (m + 1)-tupling map on X.
Suppose that T 0 is a collection of p ≥ 1 bounded external addresses that land at z 0 ; we claim that p ≤ m + 1. Indeed, for j ≥ 0, define T j . . = σ j (T 0 ). Then all filaments at addresses in T j land at f j (z 0 ). Since z 0 is not pre-periodic and its orbit does not pass through any critical points, the T j are pairwise disjoint, and σ : T j → T j+1 is an order-preserving bijection for all j. Furthermore, the T j are pairwise unlinked. That is, if j =j, then all elements of T j lie between the same two adjacent elements of Tj with respect to circular order.
This means that the set ϕ(T 0 ) is a wandering p-gon for the (m+1)-tupling map on S 1 . Kiwi [Kiw02, Theorem 1.1] proved that polynomials of degree d do not have wandering (d + 1)-gons. A combinatorial version of this result (see [BL02,Theorem B]) implies that p ≤ m + 1 as claimed.
Remark. It seems likely that one can also directly prove the absence of wandering d + 2gons for maps with at most d singular values. (Compare [ARG17] for the proof of the case d = 1, i.e. the no wandering triangles theorem for exponential maps.) This would imply that the number of filaments in Theorem 12.1 is always bounded by d+1 (assuming that z 0 is not pre-critical).

Appendix: Cyclic order of unbounded closed connected sets
In this section, suppose that A is any pairwise disjoint collection of unbounded, closed, connected subsets of C such that, for every A ∈ A, all elements of A \ {A} belong to the same connected component of C \ A. Observe that the latter condition holds, in particular, if no A ∈ A separates the plane.
The purpose of this section is to note that there is a natural cyclic order (at ∞) on A. Recall that a cyclic order is a ternary relation A ≺ B ≺ C that is cyclic, asymmetric, transitive and total [Čec69, § 5].
In our case, the relation A ≺ B ≺ C means that B lies between A and C in positive orientation. To make this precise, let us begin by defining a circular order on any finite subset of A. So suppose that A 1 , . . . , A n (n ≥ 3) are distinct elements of A. Let W j be the connected component of C \ A j that contains A i for i = j, and setÃ j . . =Ĉ \ W j . Then K . . = n j=1Ã j is a compact, connected and full set inĈ, and its complement is In other words, the simply connected domain W is the unique connected component W of C \ n j=1 A j whose boundary intersects A j for each j. We now consider the space of prime ends of W ; see [Pom92,Section 2.4]. Recall that these form a topological circle, and therefore possess a natural cyclic order. Note that the connected components of K \ {∞} are precisely theÃ j \ {∞}. It follows (e.g. as a consequence of the plane separation theorem [Why42, Theorem 3.1, Chapter VI]) that there are exactly n different accesses ζ 1 , . . . , ζ n to ∞ from W . They separate the circle of prime ends into n complementary intervals I 1 , . . . , I n , which may be labeled such that I j consists of those prime ends that can be represented by a sequence of cross-cuts both of whose endpoints belong to A j . We define the circular order of the sets A j at ∞ (in positive orientation) to be the circular order of these intervals, taken in negative orientation.
If we add a new element A n+1 of A to our collection, then it is easy to check that this does not change the definition of the circular order of A 1 , . . . , A n . Hence we do indeed obtain a well-defined circular order on all of A. Moreover, suppose thatÃ is a second collection as above, where every element ofÃ is contained in an element of A and every element of A contains exactly one element ofÃ. Then the cyclic order onÃ coincides with the corresponding order on A.
We can use this observation to define cyclic order also for pairwise disjoint collections of open unbounded domains, each of which contains exactly one homotopy class of curves to infinity. (Simply replace each domain by a representative in the mentioned homotopy class.) Furthermore, suppose that U andŨ are unbounded domains in C, that ϕ : U →Ũ is a conformal isomorphism. Also suppose that A andÃ are collections as above, whose elements are contained in U andŨ , respectively, that ϕ maps every element of A to an element ofÃ, and that all elements ofÃ arise in this manner. Then the action of ϕ on A preserves cyclic order.
Finally, let A be a pairwise disjoint collection of closed, connected sets in C * = C\{0}, and that the closure of each element of A contains both 0 and ∞. Then we can define the cyclic order at ∞ on A, by replacing each element of A by an unbounded connected subset that is closed in C, and applying the above definition. Analogously, we can define a cyclic order on A at 0. It is easy to see (again using the plane separation theorem) that both orders coincide, and depend only on A rather than any choices made in the construction.
Remark. There are some subtleties to the definition of circular order on connected sets, compared with the case of arcs to infinity which has been previously considered in the complex dynamics literature. For example, note that the assumption that the sets in A are closed is crucial. Indeed, consider the case of a Knaster bucket-handle continuum X, whose terminal point (that is, the initial point of the half-ray running through all of the endpoints of the complementary intervals of the ternary Cantor set) has been placed at ∞, and consider the collection of path-connected components of this set. Every such component is unbounded and connected, but since each component accumulates everywhere upon X, there is no sensible circular order among them.
14. Appendix: Unbounded postsingular sets As mentioned in the introduction, the Douady-Hubbard landing theorem no longer holds for polynomials with escaping singular values. It is still true that every repelling (or parabolic) periodic point is accessible from the basin of infinity, and even by a dynamic ray, if we extend this notion appropriately to the case where the ray passes through critical points; compare [EL89,LP96]). However, it is possible for the set of landing rays to be uncountable, and for none of these rays to be periodic; compare [GM93, Appendix C] and [LP96].
Let us now briefly discuss the case of transcendental entire functions f with unbounded postsingular set P(f ). When f / ∈ B, the structure of the escaping set may change dramatically within a given parameter space (compare [RS17, Appendix B]), and hence it is not clear whether questions concerning the landing of rays or filaments are even meaningful in this setting. Let us hence restrict to the case of f ∈ B.
First suppose that f has an escaping singular value. In addition to the abovementioned behaviour that occurs already for polynomials, it is also possible for a repelling periodic point to not be accessible from the escaping set at all (by hairs or filaments). Indeed, this is the case for the fixed point of the exponential map z → e z having imaginary part between 0 and π, and shows that the question of landing behaviour at periodic points becomes considerably more subtle when P(f ) is unbounded.
However, consider now the full family of exponential maps, f a : z → e z + a. Suppose that the singular value a has an unbounded orbit but does not belong to the escaping set. Then f a is criniferous. In [Rem06a], it is shown that that all periodic hairs of f a land. Conversely, every periodic point, with the exception of at most one periodic orbit, is the landing point of a periodic hair. The exceptional orbit cannot be parabolic, but it is an open question whether it can be repelling. It is shown in [Rem06a] that a plausible conjecture about parameter space of exponential maps (the "no ghost limbs conjecture") would imply that this is not the case.
Hence it is plausible that the Douady-Hubbard landing theorem remains valid for exponential maps as above, which raises the question whether the main theorem of our paper may also have an extension for functions f ∈ B with unbounded but nonescaping singular orbits. A crucial step is to ensure the landing of periodic rays (or filaments). Indeed, if periodic rays land and the function has good geometry in the sense of [RRRS11], one can show that the number of rationally invisible repelling periodic orbits is bounded by the number of free singular values [BF20], just as for the exponential family. Unfortunately, the proofs in [Rem06a] that periodic rays land use sophisticated results on the structure of the (one-dimensional) parameter space of exponential maps, and it appears that fundamentally new approaches would be required to resolve this question in full generality.