On the Lebesgue measure of the Feigenbaum Julia set

We show that the Julia set of the Feigenbaum polynomial has Hausdorff dimension less than~2 (and consequently it has zero Lebesgue measure). This solves a long-standing open question.


I
In [AL08] and [AL15], Avila  In this paper we present a new sufficient condition for the Julia set J F of F to have Hausdorff dimension dim H (J F ) less than two. Using computer-assisted means with explicit bounds on errors, we show that this condition is satisfied. Thus, we solve a long-standing open question.
Main Theorem. The Hausdorff dimension of the Julia set J F of the Feigenbaum map F is less than two. In particular, the Lebesgue measure of J F is equal to zero.
The structure of the paper is as follows. In Section 2, we briefly recall the definition of quadratic-like renormalization, describe the relevant results of Avila and Lyubich, and introduce the Feigenbaum map F. We also define the setsX n of points whose orbits intersect certain small neighborhoods of the origin, and denote byη n the relative measures ofX n . Using the Avila-Lyubich results, we conclude that in order to prove dim H (J F ) < 2 it is sufficient to show thatη n converges to 0 exponentially fast in n.
In Section 3, we describe the structure of the map F. Section 4 gives us distortion bounds for certain branches of inverse iterates of F. In Section 5, we state and prove the main result of the present paper (Theorem 5.3), which gives recursive inequalities forη n . As a result of Theorem 5.3, we obtain a sufficient condition to show dim H (J F ) < 2 (Corollary 5.4). This condition is one that can be checked by rigorous computer estimates, which we discuss in Section 6.
The authors would like to acknowledge the invaluable assistance of Misha Lyubich, who suggested the problem (as well as the collaboration) and participated in many fruitful discussions, providing constant encouragement and attention. This paper was also significantly improved by discussions with Michael Yampolsky, Sebastian van Strien and Davoud Cheraghi, to whom we are quite appreciative.

P
Recall that a quadratic-like map is a ramified covering f : U → V of degree 2, where U V are topological disks in C. We refer the reader to [DH85], [Ly97], or [McM] for a more detailed treatment. For a quadratic-like map f , its filled Julia set K f and Julia set J f are given by Let f : U → V be quadratic-like. The map f is renormalizable of period n if there there is an n > 1 and U ⊂ U for which f n : U → V = f n (U ) is a quadratic-like map with connected Julia set J , and such that the sets f i (J ) are either disjoint from J or intersect it only at the β-fixed point. In this case, f n | U is called a pre-renormalization of f ; the map R n f := Λ • f n | U • Λ −1 , where Λ is an appropriate rescaling of U , is the renormalization of f .
An infinitely renormalizable quadratic-like map f is called a Feigenbaum map if it has bounded combinatorics (that is, there is a uniform bound on the periods n of renormalization) and of bounded type (the moduli of V U are uniformly bounded).
The present paper is concerned with the Hausdorff dimension of the Julia set of the quadratic Feigenbaum polynomial f Feig (z) = z 2 + c Feig , where c Feig ≈ −1.4011551890 is the limit of the sequence of real period doubling parameters. Discovery of universality properties in the period doubling case during the 1970s by Coulet & Tresser and Feigenbaum ( [CT78], [Fe78,Fe79]) gave rise to the development of renormalization theory in dynamics. This development is well documented; for a brief overview, see [Ly99,§1.5] and the references therein, for example.
The Julia set of f Feig has been shown to be locally connected (see [HJ93,J00], also [Bu99]), although the Julia set is "hairy" in the sense that it converges to the entire plane when magnified about the critical point (see [McM,Thm. 8.7]). In contrast to the quadratic case, the Hausdorff dimension of the Julia set of a period-doubling Feigenbaum map tends to 2 as the order of the critical point tends to infinity [LS05], while the corresponding Lebesgue measure tends to zero [LS10].

T A -L
Let f be a Feigenbaum map. Let f n denote the n-th pre-renormalization of f , let J n be its Julia set, and let O( f ) be the critical orbit.
Avila and Lyubich showed the existence of domains U n ⊂ V n (called "nice domains") for which • f n (U n ) = V n ; • U n ⊃ J n ∩ O( f ); • V n+1 ⊂ U n ; • f k (∂V n ) ∩ V n = for all n, k; • A n = V n U n is "far" from O( f ); • area(A n ) area(U n ) diam(U n ) 2 diam(V n ) 2 .
The construction of U n and V n involves cutting neighborhoods of zero by equipotentials and external rays of pre-renormalizations f n of f and taking preimages under long iterates of f n .
For each n ∈ N, let X n be the set of points in U 0 that land in V n under some iterate of f , and let Y n be the set of points in A n that never return to V n under iterates of f . Introduce the quantities Theorem 2.1 (Avila-Lyubich [AL08]). Let f be a periodic point of renormalization, i.e. there is a p so that R p f = f . Then exactly one of the following is true: Lean case: η n converges to 0 exponentially fast, inf ξ n > 0, and dim H (J f ) < 2; Balanced case: η n ξ n 1 n and dim H (J f ) = 2 with area(J f ) = 0; Black Hole case: inf η n > 0, ξ n converges to 0 exponentially fast, and area(J f ) > 0.
Specific bounds determining the behavior of η n and ξ n depend on the geometry of A n and O( f ).
The proof of Theorem 2.1 relies on recursive estimates involving η n , ξ n , and the Poincaré series for f n . In the Lean case (relevant for this paper), Avila and Lyubich showed existence of a constant C > 0 which only depends on geometric bounds for U n ⊂ V n and O( f ), such that if there exists m divisible by p with η m < ξ m /C, then η n → 0 exponentially fast.
Thus, to show that for the period doubling renormalization fixed point F one has dim H (J F ) < 2, it would be sufficient to compute this constant C and find large enough m so that η m < ξ m /C. However, this task turns out to be extremely computationally complex for several reasons, including: • constructing the sets U n and V n is very technical, and it is difficult to obtain rigorous approximations of these sets computationally; • the geometry of U n and V n is complicated and U n is not compactly contained in V n , making the corresponding geometric bounds very rough; • the constant C is given implicitly; estimates show it can be very large (on the order of 10 10 ).
In our new sufficient condition for showing dim H (J F ) < 2, we overcome these difficulties by using the tiling of the plane by preimages of the upper and the lower half-planes as introduced in [Bu99]. In particular, we replace the nice domains of Avila/Lyubich by the Buff tiles containing zero on the boundary. These tiles can be approximated quite efficiently and have good geometric bounds. Moreover, the scale-invariant structure of the tiling allows us to construct explicit recursive estimates for quantities which are an analogue to η n directly, without using the Poincaré series. While our approach allows showing area(J F ) = 0 without appealing to Theorem 2.1, the results of [AL08] give the stronger result that dim H (J F ) < 2.

T -
Recall (see [Eps]) that the fixed point F of period-doubling renormalization is a solution of Cvitanović-Feigenbaum equation: where 1 λ = 2.5029 . . . is one of the Feigenbaum constants. From (2.2) we immediately obtain whenever both sides of the equation are defined.
Results of H. Epstein [Ep89,Eps] imply that there exists a domain W containing 0 such that F | W is a quadratic-like map For each n ∈ N, letX n denote the set of points z ∈ W (1) such that F k (z) ∈ W (n) for some k 0. Setη .
Thus,η n is the probability that the orbit of a point randomly chosen from W (1) with respect to Lebesgue measure will intersect W (n) . By construction,X n+1 ⊂X n for any n. Therefore,η n is non-increasing in n.
Lemma 2.4. Ifη n converges to 0 exponentially fast then η n also does.
Proof. The properties of nice domains imply that there exists n 0 such that V n+n 0 ⊂ W (n) for every n. Then X n+n 0 ⊂ λ −1X n+n 0 +1 for every n, from which the lemma follows.

S F F
For the proof of the following we refer the reader to [Eps]: Proposition 3.1. Let x 0 be the first positive preimage of 0 under F. Then λ and x 0 λ is the first positive critical point of F. A map g : U g → C is called an analytic extension of a map f : U f → C if f and g are equal on some open set. An extensionf : S ⊃ U f → C of f is called the maximal analytic extension if every analytic extension of f is a restriction off . The following crucial observation is also due to H. Epstein (cf. [Ep89,Eps]; see also [McM,§7.3

]):
Theorem 3.2. The map F has a maximal analytic extension F : W → C, where W ⊃ R is an open simply connected set which is dense in C.
Let H + = { z | Im z > 0 } be the upper half-plane, and let H − = { z | Im z < 0 } denote the lower half-plane. For a proof of the following, see [Eps] or [Bu99]: Following [Bu99], we now introduce a combinatorial partition of W.
Definition 3.5. Denote by P the set of all connected components of F −1 (C R). For each nonnegative integer n, let P (n) = { λ n P | P ∈ P } Using the Cvitanović-Feigenbaum equation (2.2), we obtain that for any non-negative integer m, the partition P (m) coincides with the set of connected components of the preimage of C R under F 2 m .
Definition 3.6. For k 0 we will refer to connected components of F −k (C R) as tiles.
In particular, an element of P (n) is a tile for any n 0, as are the half-planes H + and H − .
Hence for any tile P, there is a k 0 so that the map F k sends P bijectively onto H + or H − . Using Theorem 3.4 we obtain the following: A proof of the following can be found in [DY16].
Lemma 3.9. The map F has exactly 3 critical points the interval (0, 6), ordered as 0 Applying Theorem 3.4 and Lemma 3.9, we see that for each of the segments [0, there is exactly one tile P ∈ P in the first quadrant which contains this segment in its boundary. Notice that P has four-fold symmetry: it is invariant under multiplication by −1 and under complex conjugation.
Definition 3.11. Let c j be the non-negative real critical points of F, with 0 = c 0 < c 1 < c 2 < . . ., and for each j, let P j,I denote the tile of P in the first quadrant with [c j , c j+1 ] in its boundary.
For each K ∈ { II, III, IV }, let P j,K denote the tile in quadrant K symmetric to P j,I with respect to the imaginary axis, the origin, or the real axis, respectively. See Figure 3.12. We will sometimes omit the second index (e.g. P 2 ); in this case we will mean any of the four symmetric tiles P j,I , P j,II , P j,III , or P j,IV (or the appropriate one, depending on context). Consistent with our earlier usage for P, for any set P and any integer n 0, we let P (n) = λ n P. Proposition 3.13. The map F satisfies the following: See Figure 3.12. We refer the reader to [DY16] for the proof of Proposition 3.13. Using part iv of Proposition 3.13 we obtain Lemma 3.14. Let y 0 be such that P (1) 0,I ∩ iR = [0, y 0 ]. Then F(y 0 ) = x 0 λ , F 3 (λy 0 ) = 0, and λy 0 ∈ J F .
For the reader's convenience we list approximate values of some of the important constants.
Definition 3.15. Define each of the following quantities: • For each K ∈ { I, II, III, IV }, set H K = Int P 0,K ∪ P 1, • For n 0, we write W (n) = λ n W and H (n) = λ n H.
Remark 3.16. The restriction F : Definition 3.17. Henceforth, we define F to be the restriction of F to W. For n ∈ Z + , let F n denote the restriction of F 2 n to W (n) , that is, the n-th pre-renormalization of F.
Remark 3.18. The following observations are immediate from the properties of F described above.
i If n is odd then iii For all n ∈ N and 1 k < 2 n−1 one has F k (W (n) ) ∩ W (n) = .
Remark 3.19. For any finite piece of orbit such that x k belongs to the closure of some tile T and D F k (x 0 ) 0, one can univalently pull back T along x 0 , . . . , x k in a unique way. In particular, this is true under the condition that x i ∈ W and x i 0 for 1 i k − 1.

C
Definition 4.1. We will say that a tile Q is a copy of the As we shall see, separated copies of P (m) 0 are called this because they stay away from relevant parts of the postcritical set. Separated copies play an important role for us, in that they allow us to have control on the distortion of tiles under iteration of f .
Remark 4.2. We make the following useful observations.
i If Q is a copy of P under F k then F k : Q → P is a bijection.
ii If Q is a copy of a separated copy T then Q is separated.
iii Let T be a separated copy of P (m) 0 with F k (T) = P (m) 0 . Then for each j k, F j (T) is either a primitive or a separated copy of P (m) 0 . In particular, the set P (m) 0 is a primitive copy of itself. iv Copies of primitive copies need not be primitive or separated. For example, let T = F(P (2) 0,I ), so F 2 (T) = P (1) 0,II and T ⊂ P (1) 1,IV W (1) ; see Figure 3.12. Hence T is a primitive copy of P (1) 0,II . But P (2) 0,I ⊂ W (1) and intersects J F , so P (2) 0,I is neither a primitive nor a separated copy of P (1) 0,II .

Lemma 4.3. Let T be a copy of P (m)
, then k is divisible by 2 n−1 . Moreover, F n−1 is defined on T for = k/2 n−1 and thus Proof. Let T be as in the conditions of the lemma. Assume that k is not divisible by 2 n−1 . Let r be the remainder of k modulo 2 n−1 and let j = 2 n−1 − r. Then F j (P (m) 0 ) = F 2 n−1 s (T) for some integer s, therefore, By the nesting property (Lemma 3.8), we see that F j (P (m) 0 ) ⊂ W (n−1) and hence F j (0) ∈ W (n−1) . Remark 3.18 implies that j = 2 n−2 . But F 2 n−2 (0) = (−λ) n−2 does not belong to W (n−1) . This contradiction proves the first statement of the lemma.
To prove the second statement we show by induction that for all 0 j < one has F j n−1 (T) ⊂ W (n−1) and F j n−1 (T) ∩ J (n−1) F . The base case of the induction j = 0 is given by the conditions of the lemma. The induction step follows from the nesting property and the fact that J (n−1) Proof. Assume that there exists a separated copy T of P (m) where · denotes the integer part of a number, and set  Since T is separated there exists 0 r and 1 Taking into account that from Remark 3.18 iii we obtain that j 2 m−2 . Similarly, since we know that 2 m−1 − j 2 m−2 by again using Remark 3.18 iii . It follows that j = 2 m−2 . But (see Proposition 3.13). This contradiction finishes the proof.

K
In what follows, we use sign(P) to represent the sign of the real part of the points in P, that is, sign(P) = −1 if P is in the left half-plane, +1 otherwise.
Proposition 4.6. Let T be a primitive or a separated copy of P (m) 0 under F k with m 2. Then the inverse branch φ : Illustration of the statement of Proposition 4.6, which says (roughly) that primitive and separated copies of P (m) 0 have a definite Koebe space around them.
Proof. Assume that the statement of Proposition 4.6 is false. Let m 2 and T be either a primitive or a separated copy of P (m) Take k to be the minimal integer such that there is no such continuation.
Since F is real analytic, without loss of generality we may also assume that P (m) it is P (m) 0,I or P (m) 0,II . By Remark 3.19, F −k has a univalent analytic continuation φ on H + which can be extended to a continuous function on H + . Since k is defined to be the minimal value such that F −k does not extend, the unique critical point 0 must be in φ(H + ); moreover,  1The set C λ should not be confused with the slightly larger set By Remark 3.18 and (4.8), k is divisible by 2 m−2 . If k 2 m+1 , then since 0 ∈ φ(H + ) the nesting property tells us that we must have T ⊂ φ(H + ) ⊂ W (m+1) . In particular, T is not primitive and so must be separated. Since φ(0) must be either real or purely imaginary, and since , contradicting the hypothesis of T being separated. Hence, we must have k < 2 m+1 . This leaves us with four possibilities, each of which we rule out now.
Case k = 2 m : In this case, Case k = 2 m−1 : , we can apply Proposition 3.13 to see that T = ±λ m−1 F(P (2) i,K ) with i = 0 or i = 1 and K = I or K = IV. Since T is either primitive or separated, i 1. Again using (2.3) we obtain Case k = 3 · 2 m−1 : Here we have P (m) 0,K ⊃ φ(H + ) ⊃ P (m+1) 0,K for some K. In particular, T is not primitive. Moreover, by definition of y 0 (see Lemma 3.14), φ(H + ) will contain either y = λ m y 0 or y = −λ m y 0 which are points of J (m−1) 13. The case k = 3 · 2 m−1 from the proof of Proposition 4.6.
Case k = 2 m−2 j where j is odd: Since all the possibilities for k lead to a contradition, we have established the proposition.
Informally speaking, Proposition 4.6 tells us that the inverse branch of the iterate of F corresponding to a separated copy T admits an analytic continuation to a region with a definite Koebe space around T.

D
The Koebe Distortion Theorem (see [Du], e.g.) implies that there exists a constant C > 0 such that for any univalent function φ on Recall that P 0 is used to denote any of the tiles P 0,I , P 0,II , P 0,III , or P 0,IV (see Definition 3.11). As a consequence of Proposition 4.6, one can show the following.
Corollary 4.14. Let A and B be two measurable subsets of P 0 of positive measure and let T be a primitive or a separated copy of P (m) 0 under F k for some k 0 and m 2. Then However, we will need a slightly sharper version of this result, where the constant depends more explicitly on the set A. We devote the remainder of this section to establishing it.
Consider the slit plane C λ . From the Koebe Distortion Theorem, the function is nonzero and finite for all z and w.
Fix a univalent map φ on C λ . For every w ∈ C λ , there is a conformal isomorphism H w : C λ → D such that H w (w) = 0 and H w (w) is a positive real number. Thus, we can write φ as a composition φ = ϕ • H w , where ϕ is a univalent map on the unit disk D.
Applying the Koebe Distortion Theorem, we have The latter gives us a way to estimate C(z, w) from above.
Fix two measurable subsets A and B of P 0 of positive measure. For any z ∈ B we have Observe that for two measurable sets A ⊂ A ⊂ P 0 , we have M(A) M(A ), since for all z we have g A (z) g A (z).
As a consequence of Proposition 4.6 and the previous discussion, we obtain the following.

R
Recall that F is a quadratic-like map F : W → C ((−∞, − 1 λ ] ∪ [ 1 λ 2 , ∞)), and for a point z ∈ W, by the forward orbit of z we mean the set F k (z) k ∈ N ∪ {0} such that F k (z) is defined . Recall from Definition 3.17 that F n denotes the n-th pre-renormalization of F, that is, the restriction of F 2 n to W (n) .
Definition 5.1. Define the following (some of which we have referred to in Section 2).
• LetX n be the set of points in W (1) that eventually land in W (n) : • Denote byη n the relative measure ofX n in W (1) : 1) ) .
• Let X n,m be the set of points in W (n) whose forward orbits under F n−1 intersect W (n+m) : • Let Y n denote the set of points in W (n) whose forward orbits never return to W (n) : • Σ n is the set of points in W (n) whose forward orbits under F n−1 intersect Y n : i X n is the union of all primitive copies of P (n) 0 that lie inside W (1) , together with a countable collection of analytic curves (which form parts of the boundaries of these copies).
ii The relative measure of X n,m in W (n) is equal toη m+1 : area(X n,m ) area(W (n) ) =η m+1 .
iii By construction, X n,m ∩ Σ n,m = for all n, m.
Theorem 5.3. For every n 2 and m 1, one has η n+m M n,m area(P 0,I )η nηm+1 Before proving Theorem 5.3, let us formulate its main corollary.
Corollary 5.4. If for some n 2 one hasη n M n area(P 0,I ) < 1, then the Hausdorff dimension of J F is less than 2.
Proof. Let n be such thatη n M n area(P 0,I ) < 1. Then there is an m for whichη n M n,m area(P 0,I ) < 1; let γ =η n M n,m area(P 0,I ) for this value of n and m. By construction Σ n,k ⊂ Σ n, whenever k < , so Corollary 4.15 tells us that M n,rn+m M n,m for every r ∈ N.
As a result,η k converges to zero exponentially fast. Consequently, Lemma 2.4 tells us that the parameter η k of the Avila-Lyubich Trichotomy (Theorem 2.1) also converges to zero exponentially fast. Thus, F is in the lean case and the Hausdorff dimension of J F is less than 2.

T T 5.3
First, let us prove some auxiliary lemmas.
Lemma 5.5. Let T be a copy of P (m+n) 0 with T ⊂ W (1) X n,m . Then there is a k 0 and a primitive or separated copy Q of Let j be the minimal number for which F j (T) intersects X n,m with F j (T) ⊂ W (n) . We now show that we must have F j (T) ⊂ X n,m . Since X n,m consists of copies of W (n+m) , the nesting property tells us that either F j (T) ⊂ X n,m or it contains a copy of P (m+n) 0 under F s n−1 for some s; in this case F j (T) ∩ J (n−1) F . By Lemma 4.3, r − j is divisible by 2 n−1 . Moreover, F p n−1 (F j (T)) = P (m+n) 0 for p = (r − j)/2 n−1 and so we must have F j (T) ⊂ X n,m .
Let Q be the unique copy of P (n) 0 under F j containing T. Let us show that Q is either primitive or separated and Q X n+m . Observe that if Q ⊂X n+m , the nesting property and Lemma 4.3 would imply that either F d (Q) ⊂ P (n+m) 0 or F d (Q) ⊃ P (n+m) 0 for some d < j. If Q is primitive this is clearly impossible.
Assume that Q is not primitive. Then there exists 0 < j such that F (Q) ⊂ W (n) . Assume that is the maximal such number. If F (Q) intersects J (n−1) F then from Lemma 4.3 we obtain that c = ( j − )/2 n−1 is an integer and F c n−1 (F (Q)) = P (n) 0 . In this case F c n−1 (F (T)) = F j (T) ⊂ X n,m and so F (T) ⊂ X n,m . This contradicts the definitions of j and . Thus, F (Q) ∩ J (n−1) F = and so Q is separated.
is impossible for d < j by Lemma 4.4 and the fact that F d (Q) is either a primitive or a separated copy of P (n) 0 (see Remark 4.2 iii ). On the other hand, if F d (Q) ⊂ P (n+m) 0 for d < j then F d (T) ⊂ W (n+m) ⊂ X n,m , contradicting the definition of j. This shows that Q X n+m is nonempty, finishing the proof.
Lemma 5.6. For every primitive or separated copy Q of P (n) 0 for which Q X n+m , one has Q ∩ (Σ n,m ∪ X n,m ) = .
Proof. Assume that Q ∩ X n,m . Then by the nesting property, Q either contains or is contained in a copy T of P (n+m) 0 under F n−1 for some . If T ⊂ Q then Q intersects J n−1 F , which is impossible since Q is separated. If, on the other hand, Q ⊂ T, then by the definition ofX n+m we have Q ⊂X n+m , contradicting a hypothesis of the lemma. Thus, we must have Q ∩ X n,m = .
Now suppose that Q ∩ Σ n,m is nonempty. Let k be such that F k (Q) = P (n) 0 , and let be the minimal number such that Q ∩ F − n−1 (Y n ) . Then F s (Q) W (n) for all s > 2 n−1 , and therefore k 2 n−1 . By the nesting property, is nonempty, Q must contain a copy T of P (n) 0,L with F n−1 (T) = P (n) 0,L . Therefore Q intersects J (n−1) F , contradicting Lemma 4.4 and finishing the proof.
Let SP be the set of all primitive or separated copies Q of P (n) 0 which lie in W (1) and are such that Q X n+m .For Q ∈ SP let k be such that F k (Q) = P (n) 0 . Set Recall that Σ n,m and X n,m are symmetric with respect to the axes. Observe that F k sends X Q and Σ Q bijectively onto the intersections of X n,m and Z n,m , respectively, with one of the four quadrants.
Corollary 5.7. The sets from the collection Σ Q , X Q Q ∈ SP are pairwise disjoint.
Proof. Since Σ n,m ∩ X n,m = , we obtain immediately that Σ Q ∩ X Q = for every Q ∈ SP. Let Q 1 , Q 2 ∈ SP be distinct copies of T 1 = P (n) 0,K and T 2 = P (n) 0,L , respectively, and let k 1 , k 2 be such that F k 1 (Q 1 ) = T 1 and F k 2 (Q 2 ) = T 2 .
If k 1 = k 2 then since Q 1 and Q 2 are distinct, they must be disjoint. Assume that k 1 < k 2 and that one of two sets Σ Q 1 , X Q 1 intersects one of two sets Σ Q 2 , X Q 2 . By the nesting property (Lemma 3.8), Q 2 ⊂ Q 1 . Since F k 1 (Q 2 ) ∈ SP, we can apply Lemma 5.6 to see that F k 1 (Q 2 ) ∩ (Σ n,m ∪ X n,m ) = .
Proof of Theorem 5.3. By Lemma 5.5,X n+m is the union of all sets of the form X Q , Q ∈ SP, together with a countable set of analytic curves. Using the fact that X n,m and Σ n,m are symmetric with respect to the axes, Corollary 4.15, the definition of M n,m (Definition 5.1), and Remark 5.2 ii , we obtain area(X Q ) area(Σ Q ) M n,m area((λ −n X n,m ) ∩ P 0,I ) = M n,mηm+1 area(P 0,I ).
Since Q∈SP X Q Q∈SP Σ Q ⊂X n , we obtain area(X n+m ) M n,mηm+1 area(P 0,I ) area(X n ).

E X n Σ n
Let P (n) J denote the elements of P (n) that intersect J F . That is, P (n) Then, for each n ∈ N, let V n be the interior of the closure of the union of all tiles P ∈ P (n) J . Observe that the sets V n form a collection of nested neighborhoods of J F . Figure 3.7 shows the tiles that make up V 1 , V 2 , and V 3 in the first quadrant; V 2 is also shown in Figure 5.13.
Definition 5.9. For n 3, let W n denote the interior of the closure of the union of the copies P of H ± under F 2 n −6 with 0 ∈ P. Notice that for each n 3 there are exactly four such copies; denote by P n,K the copy in quadrant K (if the quadrant is omitted, we mean the appropriate copy). Remark 5.11. The following observations are immediate from the definitions. See Figure 5.10.
For a point z 0 J F , the next lemma gives us explicit criteria for determining a disk of points around z 0 whose orbits behave comparably. See Proof. If we assume the lemma does not hold, D 0 must contain points from ∂W (n) and ∂ W n .
First, since V 2 is a neighborhood of J F , the definition of D means it can contain no points of J F . Using D j to denote F j (D 0 ), we have D j ∩ J F = for 0 j k.
Observe also that W (n) ⊂ F −(2 n −4) (V 2 ), and hence k 2 n − 3. Applying Remark 5.11 i gives F k (∂ W n ) = F k−2 n +6 (F 2 n −6 (∂ W n )) = F k−2 n +6 (R) = R, (5.14) so we must have D k ∩ R . Further, if for some j < k we have D j intersecting both R and iR, it can be shown by induction that D = D k must contain both positive and negative real values, which is impossible.
If D j ∩ iR = for all j < k, then since D j can contain no points of J F , we must have D j ⊂ P (1) 1,I ∪ P (1) 1,II ∪ P (1) 1,III ∪ P (1) 1,IV for all j < k − 1.
This contradicts our initial hypothesis that D 0 intersects W (n) ; so for some j < k, we must have D j ∩ iR . Let s be the maximal index for which D s intersects the imaginary axis, and let x be a point in D s ∩ iR. Without loss of generality, we may take Im(x) > 0.
As noted earlier, D s cannot intersect both R and iR. Hence D s ∩ R = . Because D s contains a boundary point of F s ( W n ), by Remark 5.11 i we must have s 2 n − 7. Combining this with the fact that k 2 n − 3 yields s k − 4.
Let V 2,R be the union of closures of tiles from P (2) which intersect J F ∩ R. Lemma 3.14 implies that V 2,R ∩ iR = [−λy 0 , λy 0 ] ⊂ J F . Thus D s intersects iR outside V 2,R , so ±λy 0 D s . Since D s intersects F s (W (n) ) and hence also intersects V 2,R , we conclude that D s contains a boundary point of V 2,R .
But since k − s 4 and F 4 (∂V 2,R ∪ [λy 0 , x]) ⊂ R, it follows that D k ∩ R consists of at least two connected components; this is impossible. The contradiction finishes the proof.
Use D R (z) to denote the open disk of radius R centered at z, and recall the definition of H (1) from Definition 3.15. From Definition 5.1, recall thatX n is the set of points of W (1) that eventually land in W (n) under iterates of F, Y n are points of W (n) which never return to W (n) under non-trivial iterates of F and Σ n is the set of points in W (n) that eventually land in Y n under iterates of F n−1 .
Applying the Koebe One-Quarter Theorem together with Lemma 5.12 yields the following two useful corollaries, which enable us to estimate the size of disks which lie outsideX n or inside Σ n .
Corollary 5.15. Fix n 3. Let z ∈ H (1) J F and let k be such that F k (z) V * 2 . Suppose also that F j (z) W n for 0 j k. Let D 0 be the connected component of F −k D R (F k (z)) containing z, with R = dist(F k (z), V * 2 ). Then D 0 ∩X n = . In particular, Corollary 5.16. Fix n 3. Let z ∈ W (n) be such that w = F s n−1 (z) ∈ Y n for some s; let be such that F (w) V * 2 . Suppose also that F j (w) W n for all 0 j .
With Corollary 5.15 and Corollary 5.16 in hand, we have explicit, computable criteria for verifying that the hypotheses of Corollary 5.4 hold, showing that the Hausdorff dimension of J F is less than two.
Specifically, for some n we need to establish upper bounds on the quantitiesη n and M n . We now give algorithms to do this. These are presented assuming that F(z), F (z), W n , P 0 , etc. can be calculated exactly. In Section 6, we discuss how to account for finite precision.
To boundη n , we need an upper bound on area(X n ), since,η n = area(X n )/area(W (1) ) by definition. Exploiting the symmetry with respect to the axes allows us to work in first quadrant only.
Algorithm 5.17. Fix n 3 and r small. To compute an upper bound for area(X n ), we find a collection D n,r of disks with radius r which coverX n in the first quadrant. First, select a grid of points z so that z D r (z) covers P (1) 0 . Also, choose some upper bound K on the maximum number of iterations. For each point z in the grid run the following routine.
i For 0 < j < K, calculate F j (z) and DF j (z).
ii If F j (z) ∈ W n for some j, or if F j (z) ∈ H (1) for all j < K, add D r (z) to D n,r , and exit the routine. iii Let be such that F (z) H (1) . If dist(F (z), V * 2 ) > 4r |DF (z)|, then by Corollary 5.15, the entire disk D r (z) does not intersectX n . Otherwise, add D r (z) to D n,r .
Remark 5.18. The cover D n,r can be calculated from D n,s for r < s by replacing the grid covering P (1) 0 by one covering D n,s .
In order to bound M n = M(Σ n ) from above, we need to construct a lower bound on Σ n (i.e. a subset of Σ n ). As before, we can exploit symmetry and work only in the first quadrant.
Algorithm 5.19. Fix n 3 and r small. To get a lower bound for Σ n , we find a collection E n,r of disks D r (z) ⊂ λ −n Σ n . As in Algorithm 5.17, select a grid of points z so that P 0 ⊂ z D r (z), and fix a positive integer K. For each point z from the grid run the following routine.
i For 0 < j < K calculate F j (z) and DF j (z).
ii If F j (z) ∈ H (1) for all j < K, discard z and exit the routine.
iii Let 0 k < K be the smallest number such that F k (z) H (1) . Set w = λ n F k (z). iv For 0 < j < K, compute F j (w) and DF j (w).
v If F j (w) ∈ W n for some j, or if F j (w) ∈ H (1) for all j, discard z and exit the routine.
vi Let be such that F (w) H (1) . If dist(F (w), V * 2 ) > 4r |DF (w)| · |DF k (z)|, then by Corollary 5.16, the entire disk D rλ n (z) is contained in Σ n ; add the disk to E n,r .
Remark 5.20. In step i of Algorithm 5.17 and steps i and iv of Algorithm 5.19, we need not (and should not) compute F j (z) for all j < K. Instead, we restrict our attention to iterates z k = F j k (z) defined inductively as follows. Let j 0 = 0. Assuming z k 0, there is a maximal number m k such that z k ∈ W (m k ) ; let i k = max{0, m k − 1}. Set j k+1 = j k + 2 i k , and calculate z k+1 = F 2 i k (z k ) and |DF j k+1 (z)| = |DF j k (z)| · |DF 2 i k (z k )| via (2.3): Observe that F 2 i k is the first return map from W (m k ) to W (m k −1) (see Remark 3.18 iii ). Given n 3 we have W n ⊂ W (n−2) . Hence, for z k W n , since m k n − 1, it is not possible to have F l (z k ) ∈ W n for any 0 < l < 2 i k . Also, it can be shown that if z k ∈ H (1) , we must have F l (z k ) ∈ H (1) for 0 < l < 2 i k . Also, we should consider all k < K rather than j < K, bounding the number of evaluations of F 2 i k rather than the length of the orbit of z.

C
In this section, we discuss how we can be certain that the computation of the bounds onη 6 , M(Σ 6 ), and |P 0 | (which give us our main theorem) are sufficiently accurate, even though the bounds are necessarily computed with finite precision. We have two potential sources of error: we can not know the map F exactly (and consequently we must also approximate λ and the domains W (n) , H, etc.), and there will be some errors introduced by the approximation of exact quantities z by those representable on a computer.
The calculations for this paper were done primarily in Python with double-precision floatingpoint arithmetic satisfying the IEEE 754-2008 standard [IEEE]. In this context, only numbers of the form α × 2 e are representable, where α is a 53-bit signed (binary) integer, and the exponent e satisfies −1021 e 1024. In particular, the only real numbers which can be represented exactly are certain dyadic rationals within a (large) range. Any real number x in the representable range can be approximated by a floating-point numberx so that |x −x| < u|x|. This number u is called the unit roundoff; for IEEE double-precision u = 2 −53 ≈ 1.11 × 10 −16 .
For more details, the reader is referred to [Hi02] or [Go91], for example.
While the standard guarantees that the result of a single arithmetic operation (+, −, * , /) carried out on two (real) floating-point numbers will be correctly rounded with a relative error of at most u, we need to ensure that these small errors do not accumulate such that we lose control of the calculation. We primarily need to work with double-precision complex numbers; in [BPZ07], it is shown that the relative error for complex arithmetic is bounded by Throughout this section, we shall use the notationx to denote the approximation of the exact quantity x by one that is representable as a floating point number.
In the 1980s, Lanford [La82] calculated a high-precision approximation of F as an even polynomial of degree 80. Such approximations can be computed to precision 10 −n in a number of arithmetic operations polynomial in n [HS14], although the approximation given by Lanford is sufficient for our purposes.
Lanford gives strict error bounds on his approximation (which he calls g (0) n but we refer to asF for notational consistency). Specifically, we have the following.
Proposition 6.1. LetF be the degree 80 polynomial approximation of F from [La82]. The following upper bounds on the error apply.
Remark 6.4. As noted in Remark 5.20, we will often want to evaluate F 2 n (z) as (−λ) n F(z/λ n ). Since Corollary 6.2 gives λ and 1/λ with greater precision than can be represented as a floating-point number, we can calculate floating-point approximations of λ n and λ −n with a relative error of no more than nu 1−nu < 1.0102nu < 1.13n × 10 −16 . Remark 6.5. By the Koebe Distortion Theorem, if g : D |z| (0) → C is univalent and | | < |z|, then This observation will be useful in calculating accumulated error bounds for compositions.

A
W, H (1) , V 2 , W 6 Definition 6.6. For complex numbers a and b, let where A * is the set of complex conjugates of points of A, and −A is the negation of all points in A.
Definition 6.7. Define the following sets of floating-point numbers.
Remark 6.9. As should be apparent from the figures, it is possible to obtain much better approximations of the relevant sets than given in Definition 6.7. To do so, one can exploit the fact that the set P 0,I is fixed under the map F λ (z) = F(λz) (see [Bu99]). F λ has an attracting fixed point on ∂P 0,I at x 0 , and a repelling fixed point at x 1 ≈ 1.831 + 2.683i. This point x 1 is the unique point in ∂P 0,I ∩ ∂ W. Taking repeated preimages of the segment [0, λy 0 ] by the mapF λ yields a good approximation of P 0,I ; other pieces of P can then be approximated via preimages of P 0 . However, the polygonal sets of Definition 6.7 are much easier to obtain sharp error bounds on, and are sufficent for our purposes.
A similar calculation shows thatȞ (1) in ⊂ H (1) :F Q (1) in ⊂W in with a margin of more than .004, which by Corollary 6.3 is much greater than the necessary space since |z| < 1.75 for z ∈Q (1) in . SinceȞ in =P in ∪Q in , we know F(Ȟ (1) in ) ⊂ W and thusȞ (1) in ⊂ H (1) . To see that V 2 ⊂V 2 , recall from Remark 5.8 that V 2 = F −3 (W). We need only check thatF 3 (∂V 2 ) stays away fromW out with a sufficient margin: the distance between the boundaries of these two sets is greater than 1.5 × 10 −3 (with the closest point ofF 3 (∂V 2 ) being the image of 9.33λ 2 ).
SinceW out contains points outside the domain of definition of F, we cannot verify directly that W ⊂W out . Instead, to do this we use the properties of the map F λ (z) = F(λz) mentioned in Remark 6.9. Notice that F λ (z) can be approximated onW out with an error less than 5.1 × 10 −7 since |λz| < 1.502 onW out (see Corollary 6.3).
6.11. Showing that W ⊂W out andȞ (1) in ⊂ H (1) . F λ (P out ∪Q out ) is shaded in blue, withP out andQ out outlined in red. The image on the right is a zoom of the one on the left. Indeed, assume that P 0,I P out , and let Γ be the closure of the part of ∂P out in the first quadrant (Γ = [2.81i, 2.495 + 2.81i] ∪ [2.495 + 2.81i, 2.495]). Then ∂P 0,I must intersect Γ. Recall that ∂P 0,I is F λ -invariant with two fixed points x 0 (attracting) and x 1 (repelling); the two components of ∂P 0,I { x 0 , x 1 } are interchanged by F λ . Parameterize the two components of ∂P 0,I { x 0 , x 1 } so that the parameterizations are the two inverse branches of a function p : ∂P 0,I {x 1 } → [0, ∞) satisfying2 p(F λ (z)) = 1 2 p(z) for every z and p(x 0 ) = 0. Consider the closed set ∂P 0,I ∩ Γ; let z 1 be the point where p(z 1 ) attains the minimum on this set. As a consequence of the previous argument, either F λ (z 1 ) or F 3 λ (z 1 ) lies outsideP out (see Figure 6.11). Therefore, there exists a point z 2 ∈ P 0,I ∩ Γ with p(z 2 ) 1 2 p(z 1 ). This contradicts the definition of z 1 , and hence P 0,I ⊂P out .
To show that H (1) ⊂Ȟ (1) out it is sufficient to confirm that F λ (∂Ȟ out ) ∩ (W ∩ H + ) = . Observe that F λ (∂Ȟ out ) ∩ (W out ∩ H + ) consists of two curves, one of which belongs to T 2 and so is outside of W. The other curve γ joins the leftmost boundary ofW out and the real axis. See Figure 6.11. This curve γ must lie outside of W, since ReF λ (ž) < −0.024 for all z ∈ γ and Re(z) > 0 for z ∈ F λ (W). The error in computingF 2 λ on all ofȞ out is less than 2.3 × 10 −6 , giving a sufficient margin of error.
Using Lemma 6.10, we can calculate better approximations of H and W from below and from above. For instance, to find a more accurate approximation of P 0,I from above we use the following. Algorithm 6.14. Fix m 1 and t > 0 small. To compute an upper bound for P 0,I , we find a collection S m,t of disks with radius t which cover P 0,I . First, select a grid of points z so that z D t (z) coversP out . Include in S m,t all the disks D t (z) which intersectP in . For each remaining point z in the grid, set w = λ n z, q = λ n (0.1 + 0.1i), t 0 = λ m t and run the following routine.
i For 0 < j < 2 m , calculatew j =F j (w). Using the Koebe Theorem and Corollary 6.3, find t j such thatF j (D t 0 (w)) ⊂ D t j (w j ).
ii If D t j (w j ) contains zero, add D t (z) to S m,t and exit the routine.
iii If D t j (w j ) does not intersectW out or the quadrant to whichF j (q) belongs, discard z and exit the routine.
iv If for all 0 < j < 2 m neither of the conditions ii or iii are satisfied, then add D t (z) to S m,t . 6.3. I A 5.17 5.19 We only need to make a few straightforward substitutions in order to implement the algorithms for approximatingX 6 and Σ 6 .
We make the obvious substitution of F j byF j and DF j by DF j . When calculatingF j , we instead compute compositions ofF 2 k as described in Remark 5.20, replacing the set W (n) byW (n) in . Whenever practical, we scalež so that z/λ m lies inW (1) in , ensuring that we have |ž/λ m | < 1.17. This implies by Corollary 6.3 thatF andF agree with F and F to within 6.5 × 10 −13 and 5.6 × 10 −12 , respectively. We take K = 20.
Further, we replace P (1) 0,I byP (1) out , V 2 byV 2 , and W 6 byˇ W 6 . When checking whetherF j (ž) ∈ H (1) for all j (step ii of both algorithms, as well as Algorithm 5.19 v ), we useȞ (1) in , but otherwise we useȞ (1) out for H (1) . In calculating the orbit of a point z we keep a running bound on the accumulated total difference between the true orbit F j (z) and the aproximationF j (ž), as well as the corresponding derivatives. More specifically, when calculating the iteratesž k+1 =F j k (ž k ), we use Corollary 6.3 and Remark 6.5 to compute upper bounds δ k > |DF j k (ž) − DF j k (ž)| and k > |F j k (ž) −F j k (ž)|.
When calculating r in Algorithm 5.17 iii , dist(F (ž),V * 2 ) should be reduced by k and DF (ž k ) should be increased by δ k ; similar changes should be made in Algorithm 5.19 vi .
As long as the pointsž k remain inW (1) in for all k, we can compute compositions ofF 2 n with reasonably high precision.3 In particular, ifǧ k is a k-fold composition of such approximations and g k is the same composition of Feigenbaum maps F 2 n , we have the following worst-case bounds on k and δ k forž k ∈W (1) in . k |g k (z) −ǧ k (ž)| |g k (z) −ǧ k (ž)| 1 6.45 × 10 −13 5.59 × 10 −12 5 2.15 × 10 −10 4.45 × 10 −9 10 2.14 × 10 −7 1.43 × 10 −5 15 2.13 × 10 −4 4.57 × 10 −2 18 1.20 × 10 −2 15.14 The above bounds are the worst case; actual calculated orbits have significantly better bounds as long as they remain withinW (1) in . As noted earlier, we compute sharper bounds on the specific function values and derivatives for each pointž k , and incorporate these into our calculations in the implementation of the algorithms. This ensures that all calculated orbits are shadowed by true orbits under F.
To estimate an upper bound on M n = M A) (see Corollary 4.14) with A = (λ −n Σ n ) ∩ P 0,I , we replace the integral in the definition of g A (z) by the Riemann sum taken over the centers of the subset E n,r of A from Algorithm 5.19, that is g A (z) g E n,r (z) ≈ w k 2r 2 C 2 (z, w k ) , as w k ranges over centers of the disks in E n,r .

We then approximate M(A)
M(E n,r ) ≈ max(1/g E n,r (z j )), where z j ranges over centers of the disks from the covering S m,t of P 0,I from Algorithm 6.14 (we take m = 2, t = 2 −6 √ 2). The relative error in this approximation of M(A) can be bounded by noticing that C(z, w) C(z, z j ) C(z j , w k ) C(w k , w); for points w ∈ D r (w k ), the Koebe Theorem gives C(w k , w) 1 − r/R (1 + r/R) 3 with R = dist(w k , ∂C λ ).
In implementing Algorithm 5.17 to obtain D n,r ⊃ X n and Algorithm 5.19 for E n,r ⊂ Σ n , we take n = 6 in both cases. For D n,r we use r = 2 −17 √ 2; for E n,r , taking r = 2 −11 √ 2 is sufficient.