Quotients of torus endomorphisms and Latt\`es-type maps

We show that if an expanding Thurston map is the quotient of a torus endomorphism, then it has a parabolic orbifold and is a Latt\`es-type map.

The main purpose of this paper is to present an open problem about Thurston maps that has mystified the authors while writing [BM17].
The general underlying question is which properties of a Thurston map are of a purely topological nature, or whether more geometric or even analytic structure is required to characterize a property. Our problem is closely related to a certain classes of maps, namely Lattès and Lattès-type maps. We start with recalling some background about these classes.
Lattès maps are rational maps on the Riemann sphere C = C ∪ {∞} that are given as quotient maps of holomorphic torus endomorphisms. More precisely, a map f : C → C is a Lattès maps if and only if there exist a (non-homeomorphic and non-constant) holomorphic map A : T → T on a complex torus T and a non-constant holomorphic map Θ : T → C such that we have the following commutative diagram: Here a complex torus T is a Riemann surface whose underlying 2manifold is a 2-dimensional torus. It is then not hard to see that f is a holomorphic map and hence a rational map on C. Moreover, one can show (see Theorem 2.5) that every Lattès map f is actually a postcritically-finite rational map with a parabolic orbifold (we explain this terminology in Section 2). Verifying that a map f as in (1.1) has indeed a parabolic orbifold is the difficult part in the proof of this statement. The argument uses the holomorphicity of f in an essential way (see [Mi06] and [BM17,p. 64]).
Thurston raised the question when a map that behaves as a rational map in a certain topological way is actually "equivalent" to a rational map (see [DH93] and [BM17] for a systematic study of this point of view). In view of this, it is natural to consider topological analogs of maps as in (1.1). This means, we consider maps f : S 2 → S 2 with the property that there exists a torus endomorphism A : T 2 → T 2 (i.e., an unbranched covering map) with topological degree deg(A) ≥ 2, as well as a branched covering map Θ : T 2 → S 2 such that we have the following commutative diagram: Here, S 2 is a topological 2-sphere, and T 2 is a topological 2-torus. We use notation different from (1.1) to indicate that these are topological objects and not Riemann surfaces, meaning that the surfaces are not equipped with a conformal structure.
If a map f arises as in (1.2), then we call f a quotient of a torus endomorphism (see Definition 2.6 for a precise statement). One can show that such a map f is actually a Thurston map, i.e., a nonhomeomorphic branched covering map with a finite set of postcritical points (see Lemma 2.7). One should expect that these maps are closely related to Lattès maps. In particular, one expects a positive answer to the following question.
Problem 1.1. Does every quotient of a torus endomorphism have a parabolic orbifold?
We have repeatedly tried to tackle this problem and also consulted with various experts, but a convincing argument for a positive answer is elusive at this point. Accordingly, it seems appropriate to present a partial answer and some facts related to Problem 1.1. This is the main purpose of this paper.
To formulate our result, we first have to define Lattès-type maps. As this involves somewhat technical terminology, we will introduce these maps in an informal way for now, but will give a precise definition later in Section 2 (see Definition 2.8).
As a starting point, one notices (see the discussion at the beginning of Section 3) that by a lifting argument for each map f : S 2 → S 2 as in (1.2) one has a commutative diagram of the form: Here Θ : R 2 → S 2 is a branched covering map and A : R 2 → R 2 is an orientation-preserving homeomorphism with a suitable equivariance property with respect to the group G of deck transformations of Θ.
In the special situation of (1.3) when A is a (real) affine map on R 2 and G is a crystallographic group, one calls f a Lattès-type map (see Definition 2.8). One can show that each Lattès map is also a Lattèstype map. This immediately follows from the characterization of Lattès maps as in condition (ii) of Theorem 2.5.
Moreover, each Lattès-type map is a quotient of a torus endomorphism with a parabolic orbifold (see Proposition 2.9). In general, the converse implication is not true, but our main result provides such a converse under the assumption that the Thurston map f is expanding (see (2.3) for the precise definition).
Theorem 1.2. Let f : S 2 → S 2 be an expanding Thurston map. Then the following conditions are equivalent: (i) f is the quotient of a torus endomorphism.
(ii) f has a parabolic orbifold.
(iii) f is a Lattès-type map.
Since every quotient f of a torus endomorphism is actually a Thurston map, the previous statement gives an answer to Problem 1.1 if f is expanding.
As we already pointed out, the implication (iii)⇒(i) is known (by Proposition 2.9). The most difficult part in the proof of Theorem 1.2 is to establish the implication (i)⇒(ii). Here we cannot rely on holomorphicity as in the proof of the parabolicity of the orbifold of Lattès maps as defined in (1.1). Instead, we will use a dynamical argument based on the expansion properties of f and its associated maps.
The proof of the implication (ii)⇒(iii) relies on the fact that an expanding Thurston map with parabolic orbifold cannot have periodic critical points, which in turn implies that it is Thurston equivalent to a Lattès-type map.
One may ask to what extent some of these implications are true without the assumption that the Thurston map f is expanding. For (i)⇒(ii) this leads to the open Problem 1.1. The implication (iii)⇒(i) is still true without expansion (see Proposition 2.9). The relation between (ii) and (iii) is covered by the following statement: Let be f a Thurston map. Then f is Thurston equivalent to a Lattès-type map if and only if f has a parabolic orbifold and no periodic critical points (see Proposition 2.10). Note that Thurston maps with parabolic orbifolds and periodic critical points are also easy to classify up to Thurston equivalence: essentially, these are power maps z → z n and Chebyshev polynomials (see [BM17,Chapter 7]).
The paper is organized as follows. We review all the relevant background and preliminaries in Section 2. The proof of the implications (i)⇒(ii) and (ii)⇒(iii) in Theorem 1.2 are then given in Sections 3 and 4. We wrap up the proof of Theorem 1.2 at the end of Section 4.
Notation. When an object A is defined to be another object B, we write A := B for emphasis.
We denote by N = {1, 2, . . . } the set of natural numbers and by N 0 = {0, 1, 2, . . . } the set of natural numbers including 0. The sets of integers, real numbers, and complex numbers are denoted by Z, R and C, respectively. We let C := C ∪ {∞} be the Riemann sphere. We also consider N := N ∪ {∞}. If A ⊂ N, then lcm(A) ∈ N denotes the least common multiple of the numbers in A.
When we consider two objects A and B, and there is a natural identification between them that is clear from the context, we write A ∼ = B. For example, The cardinality of a set X is denoted by #X and the identity map on X by id X . If x n ∈ X for n ∈ N are points in X, we denote the sequence of these points by {x n } n∈N , or just by {x n } if the index set N is understood.
If f : X → X is a map and n ∈ N, then If f : X → X is a map, then preimages of a set A ⊂ X or a point p ∈ X under the n-th iterate f n are denoted by f −n (A) := {x ∈ X : f n (x) ∈ A} and f −n (p) := {x ∈ X : f n (x) = p}, respectively.
Let (X, d) be a metric space, and M ⊂ X.

Background
In this section we state some relevant definitions and collect some facts for the convenience of the reader. More details on all of these topics can be found in [BM17].
Branched covering maps. We closely follow the presentation in [BM17, Section 2.1 and Section A.6]. A surface is a connected and oriented topological 2-manifold. A surface X is a topological disk if it is homeomorphic to the unit disk D := {z ∈ C : |z| < 1}.
Let X and Y be surfaces, and f : X → Y be a continuous map. Then f is a branched covering map if for each point q ∈ Y there exists a topological disk V ⊂ Y with q ∈ V that is evenly covered by f in the following sense: for some index set I = ∅ we can write f −1 (V ) as a disjoint union of open sets U i ⊂ X such that U i contains precisely one point p i ∈ f −1 (q). Moreover, we require that for each i ∈ I there exists d i ∈ N, and orientation-preserving homeomorphisms ϕ i : U i → D and ψ i : V → D with ϕ i (p i ) = 0 and ψ i (q) = 0 such that For given f the number d i is uniquely determined by p = p i and called the local degree of f at p, denoted by deg(f, p). Our definition allows different local degrees at points in the same fiber f −1 (q).
Every branched covering map f : X → Y is surjective, open (images of open sets are open), and discrete (the preimage set of every point is discrete in X, i.e., it has no limit points in X). Every (locally orientation-preserving) covering map (see [BM17,Section A.5]) is also a branched covering map.
A critical point of a branched covering map f : X → Y is a point p ∈ X with deg(f, p) ≥ 2. A critical value is a point q ∈ Y such that the fiber f −1 (q) contains a critical point of f . The set of critical points of f is discrete in X and the set of critical values of f is discrete in Y . If f : X → Y is a branched covering map, then f is an orientationpreserving local homeomorphism near each point p ∈ X that is not a critical point of f . If X is a compact surface and f : X → X a branched covering map, then we denote by deg(f ) ∈ N the topological degree of f (see [BM17, Section 2.1 and A.4] for the precise definition and more discussion).
The following statement is useful if one has to deal with compositions of branched covering maps (see [BM17,Lemma A.16]).
Lemma 2.1 (Compositions of branched covering maps). Let X, Y , and Z be surfaces, and f : X → Z, g : Y → Z, and h : X → Y be continuous maps such that f = g • h.
(i) If g and h are branched covering maps, and Y and Z are compact, then f is also a branched covering map. Let X, Y , Z be surfaces, and h : X → Y , g : Y → Z be branched covering maps. If g • h : X → Z is also a branched covering map, then we have for all x ∈ X. We will use this multiplicativity of local degrees throughout, usually without specific reference. For the proof we refer to [BM17,Lemma A.17]. Note that there slightly stronger assumptions were used, but the proof for (2.2) remains valid without change.
Thurston maps. Throughout this paper, S 2 denotes a topological 2sphere. We assume that S 2 is equipped with a fixed orientation. To be able to use metric language, we also assume that S 2 carries a base metric that induces the given topology on S 2 . Let f : S 2 → S 2 be a branched covering map. We denote by We also say that f is postcritically-finite. If, in addition, f is defined on C and is holomorphic, then f is a postcritically-finite rational map and we call f a rational Thurston map. A periodic point of f is a point p ∈ S 2 with f n (p) = p for some n ∈ N. For more details see [BM17, Section 2.1].
Expansion. Let f : S 2 → S 2 be a Thurston map. We say that f is expanding if for some Jordan curve C ⊂ S 2 with post(f ) ⊂ C we have Here mesh(f, n, C) denotes the supremum of the diameters of components of S 2 \ f −n (C). This condition is independent of the choice of the curve C and the base metric on S 2 (see [BM17, Section 6.1]).
If f : C → C is a rational Thurston map, then it is expanding if and only if f has no periodic critical points. This is the case and only if its Julia set is the whole sphere C (see [BM17, Proposition 2.3]).
Every expanding Thurston map f : S 2 → S 2 has an associated visual metric on S 2 that induces the given topology. The metric ̺ has a associated expansion factor Λ > 1. We refer to [BM17, Chapter 8] for precise definitions. We will only need one fact about visual metrics.
Lemma 2.2. Let f : S 2 → S 2 be an expanding Thurston map, and ̺ be a visual metric for f with expansion factor Λ > 1. Then then there exists constants δ ̺ > 0 and C > 0 such that for each path α : This follows from [BM17, Lemma 8.9] and the discussion after this lemma. In other words, if we lift a path with sufficiently small diameter under f n , then the lifts shrink uniformly at an exponential rate as n → ∞.
Thurston equivalence. For Thurston maps one often considers the following notion of equivalence (see [BM17, Section 2.4] for more explanations).
Definition 2.3. Let f : S 2 → S 2 and f : S 2 → S 2 be Thurston maps. Then they are called Thurston equivalent if there exist homeomorphisms h 0 , h 1 : S 2 → S 2 that are isotopic relative to post(f ) and satisfy Here S 2 is another topological 2-sphere. Two maps f : S 2 → S 2 and f : S 2 → S 2 are called topologically conjugate if there exists a homeomorphism h : S 2 → S 2 such that h•f = f • h. It easily follows from the definitions that if two Thurston maps are topologically conjugate, then they are Thurston equivalent. The converse is true if the maps are expanding.
Theorem 2.4. Let f : S 2 → S 2 and f : S 2 → S 2 be expanding Thurston maps that are Thurston equivalent. Then they are topologically conjugate.
The orbifold associated with a Thurston map. We follow [BM17, Section 2.5]. Let f : S 2 → S 2 be a Thurston map. For a given p ∈ S 2 we set α f (p) = lcm{deg(f n , q) : q ∈ S 2 , n ∈ N, and f n (q) = p}.
Here lcm(M) ∈ N := N ∪ {∞} denotes the least common multiple of a set M ⊂ N.
Note that α f (p) = ∞ is possible. This is true if and only if p contained in a critical cycle of f , i.e., p is a fixed point and a critical point of f n for some n ∈ N. It follows that α f is finite (i.e., it does not take the value ∞) if and only if f has no periodic critical points. Note that in general an expanding Thurston map may have periodic critical points (see [BM17,Example 12.21 The function α f : S 2 → N is called the ramification function of f and O f = (S 2 , α f ) (i.e., the underlying 2-sphere equipped with this ramification function) the orbifold associated with f . The Euler characteristic of O f is defined as .
For a Thurston map f we always have Parabolic orbifolds. To give a more precise classification of Thurston maps with parabolic orbifold, we consider the postcritical points p 1 , . . . , p k , k ∈ N, of a Thurston map f labeled so that It follows from the definition of the ramification function α f that that for all p ∈ S 2 (see [BM17, Proposition 2.14]).
Lattès maps. We follow the presentation in [BM17, Chapter 3]. The definition of a Lattès map is based on the following fact (this is essentially well known; see [Mi06] and [BM17, Theorem 3.1]).
Theorem 2.5 (Characterization of Lattès maps). Let f : C → C be a map. Then the following conditions are equivalent: (i) f is a rational Thurston map that has a parabolic orbifold and no periodic critical points.
(ii) There exists a crystallographic group G, a G-equivariant holomorphic map A : C → C of the form A(z) = αz + β, where α, β ∈ C, |α| > 1, and a holomorphic map Θ : There exists a complex torus T, a holomorphic torus endomorphism A : T → T with deg(A) > 1, and a non-constant holomorphic map Θ : Here a crystallographic group G is a subgroup of the group of orientation-preserving isometries of C that acts properly discontinuously and cocompactly on C. In particular, each element g ∈ G is a map of the form g : z ∈ C → αz + β, where α, β ∈ C with |α| = 1. Note that this definition of a crystallographic group G is more restrictive than usual, because we require that all elements g ∈ G preserve orientation on C. These groups are completely classified (see [BM17,Theorem 3.7]).
A continuous map Θ : . The reason for this terminology is that under some additional assumptions (for example, when Θ is surjective and open), there exists a homeomorphism between S 2 and the quotient space R 2 /G such that Θ corresponds to the quotient map R 2 → R 2 /G (see [BM17,Corollary A.23] for a precise statement along these lines). In the literature such maps Θ are sometimes called strongly G-automorphic.
Finally, if G is a group of homeomorphisms on R 2 and A : A map f : C → C on the Riemann sphere C is a Lattès map if one, hence each, of the conditions in Theorem 2.5 are satisfied. Such a map is always expanding (see [BM17, Proposition 2.3]). Note that condition (iii) in this theorem was how we introduced Lattès maps in the introduction.
In the following, for h ∈ R 2 ∼ = C we denote by τ h : R 2 → R 2 the translation defined as The subgroup of all translations in a crystallographic G is denoted by G tr . One can show that for each crystallographic G there exists a rank-2 lattice Γ ⊂ R 2 such that G tr = {τ γ : γ ∈ Γ}. In particular, G tr also acts cocompactly and properly discontinuously on R 2 . Moreover, the quotient space T = C/G tr ∼ = R 2 /Γ is a torus carrying a natural complex structure, and it is hence a complex torus.
If f is a Lattès map, then one can always find A, A, G, Θ as in Theorem 2.5 such that we have the following commutative diagram (see [BM17,(3.10)]): Here π : C → C/G tr = T is the quotient map which is the universal covering map of the torus T. As we will see, a topological analog of (2.7) will be the starting point for the proof of Theorem 1.2.
Quotients of torus endomorphisms. We first record a precise definition for a quotient of a torus endomorphism. As before, we will denote by T 2 a 2-dimensional topological torus. We call a branched covering map A : T 2 → T 2 a torus endomorphism. It easily follows from the Riemann-Hurwitz formula that A actually cannot have critical points, and so must be a (locally orientation-preserving) covering maps. We can now give a precise definition of the most important concept in his paper.
Definition 2.6 (Quotients of torus endomorphisms). Let f : S 2 → S 2 be a map on a 2-sphere S 2 such that there exists a torus endomorphism A : T 2 → T 2 with deg(A) ≥ 2, and a branched covering map Θ : Then f is called a quotient of a torus endomorphism.
In this case, we have a commutative diagram as in (1.2). The following statement summarizes some facts about these maps (see [BM17, Lemmas 3.12 and 3.13]). (iii) f has a parabolic orbifold if and only if for all x, y ∈ T 2 with Θ(x) = Θ(y).
This last parabolicity criterion will be important for us. The condition stipulates that the local degree deg(Θ, ·) is constant on the fiber for all x, y ∈ R 2 and λ ∈ R. In other words, an R-linear map L : R 2 → R 2 is a linear map on R 2 considered as a vector space over R. We write det(L) ∈ R for the determinant of L.
The map L A is uniquely determined by A and called the linear part of A.
We can now give a precise definition of a Lattès-type map.
Definition 2.8 (Lattès-type maps). Let f : S 2 → S 2 be a map such that there exists a crystallographic group G, an affine map A : R 2 → R 2 with det(L A ) > 1 that is G-equivariant, and a branched covering map Θ : Note that then we have a commutative diagram as in (1.3). It follows from condition (ii) in Theorem 2.5 that every Lattès map is also of Lattès-type.
Lattès-type maps are natural non-holomorphic analogs of Lattès maps. In this context we usually write R 2 instead of C for the plane, to emphasize that we do not rely on a complex structure.
If f : S 2 → S 2 is a Lattès-type map, G is a crystallographic group, and Θ : R 2 → S 2 is induced by G as in Definition 2.8, then G is necessarily of non-torus type, meaning that G is not isomorphic to the rank-2 lattice Z 2 . This implies that there is a natural identification R 2 /G ∼ = S 2 of the quotient space R 2 /G with the underlying 2-sphere S 2 . Under this identification Θ corresponds to the quotient map R 2 → R 2 /G (see [BM17,Section 3.4]).
In the the following, we summarize some facts about these maps. Note first that if f is a Lattès-type map, and A is as in Definition 2.8 with its linear part L A , then det(L A ) = deg(f ) ≥ 2 (see [BM17,Lemma 3.16]). This is underlying reason for the requirement det(L A ) > 1 in Definition 2.8.
Some of the relations between Lattès-type maps, quotients of torus endomorphisms, and Thurston maps with parabolic orbifold are covered by the following two results. Only the the last signature leads to maps that are genuinely different from Lattès maps.
A Lattès-type map is not necessarily expanding as a Thurston map (see [BM17,Example 6.15]). In order to record a criterion for this, we call an R-linear map L : R 2 → R 2 expanding if |λ| > 1 for each of the (possibly complex) roots λ of the characteristic polynomial P L (z) := det(L − z id R 2 ) of L. Lattices and tori. We quickly review some facts about lattices and tori (see [BM17,Section A.8] for more details).
A lattice Γ ⊂ R 2 is a non-trivial discrete subgroup of R 2 (considered as a group with vector addition). The rank of a lattice is the dimension of the subspace of R 2 (considered as a real vector space) spanned by the elements in Γ. Here we are only interested in rank-2 lattices Γ, i.e., lattices Γ ⊂ R 2 that span R 2 .
If Γ ⊂ R 2 is a rank-2 lattice, then the quotient space R 2 /Γ (equipped with the quotient topology) is a 2-dimensional torus T 2 , and the quotient map π : R 2 → T 2 = R 2 /Γ is a covering map. The lattice translations τ γ , γ ∈ Γ, are deck transformations of the quotient map π and so π = π • τ γ for γ ∈ Γ. Actually, every deck transformation of π has this form (see [BM17,Lemma A.25 Every topological torus can be represented in the form R 2 /Γ up to an orientation-preserving homeomorphism. In the following lemma we collect various statements that are used later.
(i) If A : T 2 → T 2 is a torus endomorphism, then A can be lifted to a homeomorphism on R 2 , i.e., there exists a homeomorphism A : R 2 → R 2 such that A • π = π • A. The homeomorphism A is orientation-preserving, and unique up to postcomposition with a translation τ γ , γ ∈ Γ. (ii) If A : T 2 → T 2 is a torus endomorphism, then there exists a unique invertible R-linear map L : R 2 → R 2 with L(Γ) ⊂ Γ such that for every lift A as in (i) we have This is part of [BM17, Lemma A.25]. Note that there it was not explicitly stated that the linear map L in (ii) is invertible. This was addressed in the proof though: one observes that the inclusion L(Γ) ⊂ Γ and the relation A • τ γ • A −1 = τ L(γ) for γ ∈ Γ imply that the map γ ∈ Γ → L(γ) ∈ Γ is injective. So L : Γ → Γ is an injective group homomorphism. Since Γ is a rank-2 lattice, L must be invertible as an R-linear map on R 2 .
One can identify Γ with the fundamental group of T 2 . Then the linear map L is essentially the map on the fundamental group of T 2 induced by A. For a careful explanation of this, see the discussion after [BM17, Lemma A.25].
Lifts by branched covering maps. Since a Thurston map is a branched covering map, we need slight variants of the standard lifting theorems for unbranched covering maps. We list a useful uniqueness result (this is essentially [BM17, Lemma A.19 (i)]).
Lemma 2.14. Let X, Y , and Z be surfaces and f : X → Y be a branched covering map. Suppose g 1 , g 2 : Z → X are continuous and discrete maps such that f • g 1 = f • g 2 . If there there exits a point z 0 ∈ Z such that p := g 1 (z 0 ) = g 2 (z 0 ) and f (p) ∈ Y \ f (crit(f )), then g 1 = g 2 .
Note that g 1 and g 2 can be considered as lifts of the map h := f •g 1 = f • g 2 under f . So this is really a uniqueness statement for lifts under f .
The condition y := f (p) ∈ Y \ f (crit(f )) is the same as the requirement that y is not a critical value of f , or equivalently, that the fiber f −1 (y) contains no critical point of f . We will apply it in the case when f : S 2 → S 2 is a Thurston map. Then this condition is satisfied if p ∈ S 2 \ f −1 (post(f )), because this implies that f (p) ∈ S 2 \ post(f ) ⊂ S 2 \ f (crit(f )).

Parabolicity of the orbifold
In this section we will prove the implication (i) ⇒ (ii) in Theorem 1.2. Throughout the section, we assume that f : S 2 → S 2 is a given quotient of a torus endomorphism that is expanding as a Thurston map (see Lemma 2.7 (i)). Then there exists a torus T 2 , and maps A : T 2 → T 2 and Θ : T 2 → S 2 as in (1.2). We can identify T 2 with a quotient R 2 /Γ, where Γ ⊂ R 2 is a rank-2 lattice, and we obtain a quotient map π : R 2 → T 2 ∼ = R 2 /Γ. The map A lifts to an orientation-preserving homeomorphism A : R 2 → R 2 such that π • A = A • π (this is standard and explicitly formulated Lemma 2.13 (i)). We define Θ = Θ • π. This is a branched covering map, since π is a covering map and Θ is a branched covering map (see Lemma 2.1 (i)). This leads to the following commutative diagram: We denote by G the group of all deck transformations of Θ, i.e., the group of all homeomorphisms g : R 2 → R 2 such that Θ•g = Θ. Since Θ preserves orientation, the same is true for each homeomorphism g ∈ G.
Recall that τ h for h ∈ R 2 denotes the translation on R 2 given by τ h (u) = u + h for u ∈ R 2 . Then π • τ γ = π for γ ∈ Γ. This implies that all lattice translations τ γ , γ ∈ Γ, belong to G; indeed, for γ ∈ Γ we have Our goal is to show that f has a parabolic orbifold. To do so, we want to apply Lemma 2.7 (iii). Essentially, we have to show that the group G of deck transformations acts transitively on each fiber of Θ, i.e., on each of the sets Θ −1 (p), p ∈ S 2 . This means we have to analyze some properties of the fixed map Θ in (3.1). Note that the diagram (3.1) remains valid with the same map Θ, if we replace A with A ′ = τ γ • A for any γ ∈ Γ. We can also replace f, A, A with iterates f n , A n , A n , respectively. We will make such replacements whenever this is convenient. The map A induces an invertible linear map L : R 2 → R 2 such that L(Γ) ⊂ Γ and for all γ ∈ Γ (see Lemma 2.13 (ii)). As we mentioned, this map L can be viewed as the homomorphism induced by A on the fundamental group on T 2 (see the discussion after Lemma 2.13). If we use (3.2) repeatedly, then we see that for all n ∈ N and γ ∈ Γ.
If x ∈ R 2 and γ ∈ Γ, then (3.2) implies that Since the lattice translations τ γ , γ ∈ Γ, act cocompactly on R 2 , it follows that there exists a constant C 0 ≥ 0 such that So the maps A and L agree "coarsely" on large scales. Before we go into more details, we outline the ensuing argument. Since f is an expanding Thurston map, we first want to translate this expansion property of f into expansion properties for the above maps A and L. In particular, L is an expanding linear map (Corollary 3.3). As we already mentioned, in order to prove that f has a parabolic orbifold (see Proposition 3.7). we have to show that G acts transitively on the fibers Θ −1 (p), p ∈ S 2 (see Lemma 3.6). In [Mi06] one can find related considerations for Lattès maps. There the holomorphicity of the underlying maps is crucially used. Here we will instead give a dynamical argument relying on the expansion property of A. We now proceed to establishing the details.
Expansion properties. We start with expansion properties of A. Actually, it is easier to formulate and prove contraction properties of A −1 . In the following all metric notions on R 2 refer to the Euclidean metric and all metric notions on S 2 to a fixed visual metric ̺ for f with expansion factor Λ > 1.
Lemma 3.1. Let Θ : R 2 → S 2 be a map as (3.1). Then the following statements are true: (i) For each ǫ > 0 there exists δ > 0 such that for all x, y ∈ R 2 we have The statement and its proof are a small modification of the similar statement [BM17, Lemma 6.14].
Proof. (i) The assertion is that Θ is uniformly continuous on R 2 . Essentially, this follows from the fact that the group G of deck transformations of Θ contains the subgroup G ′ := {τ γ : γ ∈ Γ} of all lattice translations and that this subgroup G ′ acts isometrically and cocompactly on R 2 .
In particular, we can find a compact fundamental domain F ⊂ R 2 for the action of G ′ on R 2 . Now suppose x, y ∈ R 2 and |x − y| is small. Then there exists g ∈ G ′ such that g(x) ∈ F . If |x − y| is small enough, as we may assume, then g(x), g(y) ∈ U, where U is a fixed compact neighborhood of F . Since Θ is uniformly continuous on U, and |g(x) − g(y)| = |x − y|, it follows that ̺(Θ(x), Θ(y)) = ̺(Θ(g(x)), Θ(g(y)) is small only depending on |x−y|. The uniform continuity of Θ follows.
(ii) We argue by contradiction and assume that the statement is false. Then there exist connected sets K n ⊂ R 2 such diam ̺ (Θ(K n )) → 0 as n → ∞, but diam(K n ) ≥ ǫ 0 for n ∈ N, where ǫ 0 > 0.
We pick a point x n ∈ K n for n ∈ N. If we replace each set K n with its image K ′ n = g n (K n ) for suitable g n ∈ G ′ , where again G ′ := {τ γ : γ ∈ Γ} and pass to a subsequence if necessary, then we may assume that the sequence {x n } converges, say x n → x ∈ R 2 as n → ∞. Note that diam(K ′ n ) = diam(K n ) and Θ(K ′ n ) = Θ(K n ). Let p := Θ(x). Since Θ : R 2 → S 2 is a branched covering map, the set Θ −1 (p) is discrete in R 2 and consists of isolated points. In particular, x ∈ Θ −1 (p) is an isolated point of Θ −1 (p) and so there exists a constant m > 0 such that |y − x| ≥ m whenever x, y ∈ Θ −1 (p) and x = y.
Pick a constant c with 0 < c < min{ǫ 0 /2, m}. The set K n is connected, and has diameter diam(K n ) ≥ ǫ 0 > 2c. Hence K n cannot be contained in the disk {z ∈ R 2 : |z − x n | < c}, and so it meets the circle {z ∈ R 2 : |z − x n | = c}. It follows that there exists a point y n ∈ K n with |x n − y n | = c. By passing to another subsequence if necessary, we may assume that the sequence {y n } converges, say y n → y ∈ R 2 as n → ∞. Then |x − y| = c < m. Note that Θ(x n ), Θ(y n ) ∈ Θ(K n ) for n ∈ N, and diam ̺ (Θ(K n )) → 0 as n → ∞. So and x, y ∈ Θ −1 (p). Since |x − y| = c > 0, we have x = y. Then x and y are two distinct points in Θ −1 (p) with |x − y| = c < m. This contradicts the choice of m, and the statement follows.
After this preparation, we now turn to the contraction properties of the map A −1 .
Lemma 3.2. Let the map A : R 2 → R 2 be as in (3.1). If ǫ 1 , ǫ 2 > 0, then there exists n 0 ∈ N such that for all x, y ∈ R 2 and n ∈ N with n ≥ n 0 .
The lemma essentially says that high iterates of A −1 shrink distances that are not too small by an arbitrarily small factor. Conversely, by applying the statement to x = A n (u) and y = A n (v) for u, v ∈ R 2 , we see that sufficiently high iterates of A expand distances that are not too small by an arbitrarily large factor.
Now suppose x, y ∈ R 2 are arbitrary, and let S be the line segment joining x and y. Then S can be broken up into N ∈ N line segments of diameter < δ where N ≤ |x − y|/δ + 1. We can apply the previous considerations for each of these smaller (parametrized) line segments in the role of β. By what we have seen, for n ≥ n 0 each of these smaller line segments has an image under A −n of diameter < δ by (3.7). Since the concatenation of these N image paths is the path A −n (S) connecting A −n (x) and A −n (y), we conclude for n ≥ n 0 , as desired.
Recall that an R-linear map L : R 2 → R 2 is called expanding if |λ| > 1 for each of the (possibly complex) roots λ of the characteristic polynomial P L (z) = det(L − z id R 2 ) of L.
Corollary 3.3. Suppose the linear map L : R 2 → R 2 is as in (3.2). Then L is expanding.
Proof. We argue by contradiction and assume that L is not expanding. Choosing ǫ 1 = ǫ 2 = 1/2 in Lemma 3.2, we can find a number n ∈ N such that for all u, v ∈ R 2 . In other words, under A n large distances are roughly expanded by the factor 2. Let λ 1 , λ 2 ∈ C be be the two (possibly identical) roots of the characteristic polynomial P (z) = det(L − z id R 2 ) of L. We may assume |λ 1 | ≤ |λ 2 |. Since P has real coefficients, we have λ 2 = λ 1 if λ 1 is not real. Moreover, λ 1 λ 2 = det(L) = deg(A) = deg(f ) ≥ 2 (see Lemma 2.13 (iii) and Lemma 2.7 (i)). So the only possibility that L can fail to be expanding is if λ 1 is real and |λ 1 | ≤ 1. Then there exists e ∈ R 2 , e = 0, such that L(e) = λ 1 e.
If Γ is the lattice chosen as in the beginning of this section, then (3.3) shows that for all x ∈ R 2 . Since the lattice translations τ γ , γ ∈ Γ, act cocompactly on R 2 , this implies that there exists a constant C ≥ 0 such that for all u ∈ R 2 . Combining this with (3.9), we see that for all u, v ∈ ℓ. So under the map A n distances along ℓ are expanded by at most by an additive term. This is irreconcilable with (3.8), and we get a contradiction. The statement follows.
We record the following consequence. Moreover, if U is bounded in addition, then U ⊂ A n (U) for all sufficiently large n ∈ N.
Proof. Let U be a neighborhood of x. Then there exists ǫ > 0 such that B := {z ∈ R 2 : |z − x| < ǫ} ⊂ U. If y ∈ R 2 is arbitrary, then Lemma 3.2 implies that |A −n (y) − x| = |A −n (y) − A −n (x)| is arbitrarily small for n ∈ N sufficiently large. Hence there exist n ∈ N such that A −n (y) ∈ B ⊂ U, and so y ∈ A n (U). It follows that R 2 = n∈N 0 A n (U). If U is bounded, then there exists R > 0 such that U ⊂ B ′ := {z ∈ R 2 : |z − x| < R}. Applying Lemma 3.2 for ǫ 1 = ǫ/(2R) and ǫ 2 = ǫ/2, we see that for all sufficiently large n ∈ N. Hence U ⊂ A n (U) for all large n.
Transitive action on fibers. Next, we will show that the group G of deck transformations of the map Θ as in (3.1) acts transitively on each fiber Θ −1 (p), p ∈ S 2 . We first show that this is true in a special case.
Proof. Let the points p ∈ S 2 , x, y ∈ R 2 ,x,ȳ ∈ T 2 be given as in the statement. In particular, we assume that A(x) =x. Note that and so x, A(x) ∈ π −1 (x). This means that Note that then A 0 (x) = A(x) + γ 0 = x, and so A 0 has the fixed point x. Recall from the discussion following (3.1) that we may replace A in this diagram with A 0 (while all the other maps remain the same). In other words, we are reduced to the case when A(x) = x in addition to our other hypotheses.
We now consider the cases A(ȳ) =ȳ and A(ȳ) =x separately. Case I: A(ȳ) =x. This is the easy case. Note that π(A(y)) = (π • A)(y) = (A • π)(y) = A(y) = x, and so A(x), A(y) ∈ π −1 (x). This implies that we can find γ ∈ Γ with A(y) − A(x) = γ. Then Thus g := A −1 • τ γ • A is a homeomorphism on R 2 with g(x) = y. We want to show that g ∈ G, meaning we need to verify that Θ • g = Θ. We know that τ γ ∈ G. Using f • Θ = Θ • A from (3.1), we obtain We now apply Lemma 2.14 for the branched covering maps Θ and Θ•g. Note that (Θ • g)(x) = Θ(y) = Θ(x) = p and p ∈ S 2 \ f −1 (post(f )). It follows that Θ = Θ • g. We proved the statement in Case I. Case II:x andȳ are fixed points of A. This case is much harder, since there is no translation τ γ with γ ∈ Γ that maps A(x) to A(y). To construct a deck transformation of Θ as in the statement, we first show that we can obtain a local one. Claim 1. There is a homeomorphismg : U → V between bounded and connected open neighborhoods U and V of x and y, respectively, with Θ •g = Θ on U.
To prove this, we note that our assumption p ∈ S 2 \ f −1 (post(f )) ⊂ S 2 \ post(f ) implies that Θ, and hence also Θ, has no critical point over p, because post(f ) = Θ(crit(Θ)) (see Lemma 2.7 (ii)). In particular, Θ is a local homeomorphism near both points x, y ∈ Θ −1 (p). This implies that there exist bounded and connected open neighborhoods U ⊂ R 2 of x, V ⊂ R 2 of y, and W ⊂ S 2 of p such that Θ|U : U → W and Θ|V : V → W are homeomorphisms. Definingg := (Θ|V ) −1 • (Θ|U) on U gives the desired map, proving Claim 1. Now the idea is to extendg to a deck transformation on R 2 by using the dynamics of A near its fixed point x. By Corollary 3.4, we know that U ⊂ A n (U) for all sufficiently large n ∈ N. Replacing A with such an iterate A n (and consequently f with f n , and A with A n ), we may assume that U ⊂ A(U). Note then we still have A(x) = x and A(y) = y for the new map A. We make the assumption U ⊂ A(U) from now on. We know that A(y) = y, but, in general, the point y will not be a fixed point of A. We have π(A(y)) = (π • A)(y) = (A • π)(y) = A(y) = y, and so y, A(y) ∈ π −1 (y). It follows that A(y) = y + γ for some γ ∈ Γ. So if we defineÃ(u) = A(u) − γ = (τ −γ • A)(u) for u ∈ R 2 , theñ A(y) = y.
Define U n = A n (U) for n ∈ N 0 . The sets U n are connected open sets containing x. We have U n ⊂ U n+1 for n ∈ N 0 , and n∈N 0 U n = R 2 . The last fact follows from Corollary 3.4.
We now define homeomorphisms g n mapping U n into R 2 recursively, by setting g 0 :=g on U 0 = U, and for n ∈ N 0 . Note that this makes sense, because A −1 (U n+1 ) = U n . This definition implies that One verifies by induction that Θ • g n = Θ on U n for all n ∈ N. Indeed, this is true for n = 0 by definition of g 0 =g. If it is true for n ∈ N 0 , then it is also true for n + 1, because By induction one also shows that g n (x) = y for all n ∈ N 0 . Indeed, g 0 (x) =g(x) = y, and if this is true for n ∈ N 0 , then it is also true for n + 1, because Claim 2. We have g n+1 |U n = g n for all n ∈ N 0 . To see this, we want to apply Lemma 2.14 to the branched covering map Θ : R 2 → S 2 , and the maps g n and g n+1 |U n . We know that on the connected open set U n . Moreover, g n (x) = y = g n+1 (x) and the point y which lies in the fiber over p = Θ(y) not containing any critical point of Θ. Claim 2 follows.
So each homeomorphism g n extends the previous one. Since the sets U n , n ∈ N 0 , exhaust R 2 , there exists a unique map g : R 2 → R 2 such that g|U n = g n for all n ∈ N 0 . It is clear that g is continuous and injective, because the maps g n have these properties. Moreover, it is clear that g(x) = y and Θ = Θ • g on R 2 .
To finish the proof in the Case II at hand, it remains to show that g : R 2 → R 2 is surjective. To do this, let us shift our attention to the images g(U n ).
Claim 3. We have g(U n ) = g n (U n ) =Ã n (V ) for all n ∈ N 0 . Recall that V =g(U) was the neighborhood of y defined in Claim 1. Thus Claim 3 is true for n = 0, since Moreover, if it is true for n ∈ N 0 , then it is also true for n + 1, because proving Claim 3.
Recall that in (3.1) we can replace A withÃ = τ −γ • A. Since the mapÃ has the fixed point y, we can apply Corollary 3.4 toÃ; so the images of the neighborhood V of y under iterates ofÃ will exhaust R 2 , i.e., we have n∈NÃ n (V ) = R 2 . Using Claim 3, we conclude that g is surjective. This finishes the proof of Case II. The statement follows.
We now show transitivity of the action of G on the fibers of Θ.
Proof. We will first show the existence of a point p 0 ∈ S 2 \ post(f ) for which G acts transitively on the fiber Θ −1 (p 0 ), and then deal with the general case.
The periodic points of f are dense in S 2 (see [BM17, Corollary 9.2]); in particular, we can find a periodic point p 0 ∈ S 2 \ post(f ). By replacing f with suitable iterates f n (and A with A n ), we may assume that p 0 is a fixed point of f . Note that then we still have p 0 ∈ S 2 \ post(f ), because the postcritical sets of a Thurston map and any of its iterates agree.
Since p 0 is a fixed point of f , the map A sends the set Θ −1 (p 0 ) into itself. It follows that each point in Θ −1 (p 0 ) is either a periodic point of A or is mapped to a periodic point of A under all sufficiently high iterates of A. If we again replace f and A with carefully chosen iterates, we can reduce ourselves to the following situation: p 0 ∈ S 2 \ post(f ) is a fixed point of f , the set Θ −1 (p 0 ) contains at least one fixed point of A, and each point in Θ −1 (p 0 ) is either a fixed point of A or mapped to a fixed point by A. Moreover, since p 0 is a fixed point of f , and p 0 ∈ S 2 \ post(f ), we have p 0 ∈ S 2 \ f −1 (post(f )). We pick a fixed pointx ∈ Θ −1 (p 0 ) of A, and a point x ∈ R 2 with π(x) =x. Then x ∈ Θ −1 (p 0 ). Now let y ∈ Θ −1 (p 0 ) be arbitrary. Thenȳ := π(y) ∈ Θ −1 (p 0 ). By our choices,ȳ is a fixed point of A or z := A(y) is a fixed point of A.
In the first case, there exists g ∈ G such that g(x) = y by the first part of Lemma 3.5.
In the second case, z is a fixed point of A. Pick z ∈ R 2 such that π(z) = z. Then z ∈ Θ −1 (p 0 ), and again there exists g 1 ∈ G with g 1 (x) = z. Moreover, by Lemma 3.5 there exists g 2 ∈ G such that g 2 (z) = y. Then g := g 2 • g 1 ∈ G and g(x) = y.
This shows that G acts transitively on the fiber Θ −1 (p).
We are now ready to prove the implication (i) ⇒ (ii) in Theorem 1.2.
Proposition 3.7. Suppose f : S 2 → S 2 is an expanding Thurston map that is a quotient of a torus endomorphism. Then f has a parabolic orbifold.
Proof. We can use all the previous considerations for the maps as in (3.1) and the deck transformation group G of Θ.
So the local degree of Θ in each fiber Θ −1 (p), p ∈ S 2 , is constant. Now Lemma 2.7 (iii) implies that f has a parabolic orbifold.

From parabolic orbifolds to Lattès-type maps
In this section we will prove the implication (ii) ⇒ (iii) in Theorem 1.2. We first establish an auxiliary fact that helps us in identifying Lattès-type maps.
Lemma 4.1. Let f : S 2 → S 2 be a map that is topologically conjugate to a Lattès-type map. Then f itself is Lattès-type map.
As we will see, the proof is fairly straightforward. There is a subtlety though that arises from the fact that by definition a branched covering map is orientation-preserving (see (2.1)), but a homeomorphism as in the definition of topological conjugacy may actually reverse orientation. In order to address this, we have to compensate by complex conjugation on C in a suitable way.
In the following, we denote by σ : C → C, σ(z) = z for z ∈ C, complex conjugation on C. Note that this is an R-linear orientationreversing isometry on C with σ −1 = σ.
Proof of Lemma 4.1. Suppose f : S 2 → S 2 is a map that is topological conjugate to a Lattès-type map f : S 2 → S 2 . Here S 2 is another topologically 2-sphere. Both S 2 and S 2 carry some fixed orientations.
According to Definition 2.8, there exists a crystallographic group G acting on C ∼ = R 2 , an G-equivariant (real) affine map A : C → C, and a branched covering map Θ : C → S 2 induced by G with f • Θ = Θ • A. Moreover, the linear part L A of A satisfies det(L A ) > 1.
Since f and f are topologically conjugate, there exists a homeomorphism h : S 2 → S 2 such that f • h = h • f . We now have to distinguish two cases according to whether the homeomorphism h : S 2 → S 2 preserves or reverses orientation. We will treat the latter, slightly more difficult case in detail, and then comment on the small modifications for the former case.
So we now assume that h : S 2 → S 2 is orientation-reversing. We define Θ := h • Θ • σ. Then Θ : C → S 2 is a branched covering map, as easily follows from the definitions and the fact that Θ : C → S 2 is a branched covering map. Here it is important that in the definition of Θ we compensate postcomposition of Θ with the orientation-reversing homeomorphism h by precomposition with the orientation-reversing homeomorphism σ to make Θ orientation-preserving.
We conjugate everything else by σ. More precisely, we define G := {σ • g • σ : g ∈ G}. It is clear G is a crystallographic group on C and that Θ is induced by G. Moreover, we let A := σ • A • σ. Then A is a (real) affine map on C ∼ = R 2 that is G-equivariant. For the linear part L A of A we have L A = σ • L A • σ, and so det(L A ) = det(L A ) > 1.
We can summarize the relations of all the maps considered in the following diagram, which is obviously commutative: It follows that f is a Lattès-type map.
If h is orientation-preserving, the map σ is not needed in the previous argument. Formally, we can just replace σ with the identity map on C. So f is also a Lattès-type map in this case. The statement follows.
After this preparation, we now prove the implication (ii) ⇒ (iii) in Theorem 1.2.
Proposition 4.2. Let f : S 2 → S 2 be an expanding Thurston map with a parabolic orbifold. Then f is a Lattès-type map.
The fact that f , or equivalently f 2 , is expanding (see [BM17, Lemma [Lemma 6.5]), rules out the signatures (∞, ∞) and (2, 2, ∞). Indeed, if a Thurston map f has one of these signatures, then f or f 2 is a Thurston polynomial (i.e., there is a point that is completely invariant