The Maximal Entropy Measure of Fatou Boundaries

We look at the maximal entropy (MME) measure of the boundaries of connected components of the Fatou set of a rational map of degree greater than or equal to 2. We show that if there are infinitely many Fatou components, and if either the Julia set is disconnected or the map is hyperbolic, then there can be at most one Fatou component whose boundary has positive MME measure. We also replace hyperbolicity by the more general hypothesis of geometric finiteness.


Introduction
Let R : C → C be a rational map of degree d ≥ 2, defined on the Riemann sphere C, F its Fatou set, J its Julia set, and λ the unique maximal entropy measure (MME) on J (i.e., h λ (R) = log d). Recall that F and J are invariant under R, R preserves λ and acts ergodically on (J , B, λ), (1.1) where B consists of the Borel subsets of J , and supp λ = J . See [6], [4], or §5.4 of [10] for (1.1)-(1.2). We want to study λ(∂O), when O is a connected component of F, and in particular to understand when we can say λ(∂O) = 0. The results in this paper hold for any Borel probability measure µ satisfying (1.1) and (1.2) It is well known that for many rational maps there are measures in addition to the MME measure λ that have these properties. We focus on λ to complement an earlier result by the authors on the construction of the MME. In the jointly written appendix of [5], the authors showed that the backward random iteration method works for drawing the Julia set of any rational map. They proved the result by showing the associated delta mass measures, averaged along almost any randomly chosen backward path, converge to λ. It is observed that certain details of Julia sets have a fuzzy appearance for many maps when this algorithm is used, and the results of this paper provide an explanation of this observation.
To get started, given a rational map R and a component O ⊂ F, Sullivan's Non-wandering Theorem implies there exist m ∈ N and a collection O 1 , . . . , O k of mutually disjoint connected components of F such that and Our first goal is to establish the following.  5) or The following general result will be useful in the proof of Theorem 1.
Proof of Theorem 1.1. Lemma 1.2 applies to E = ∂O 1 ∪ · · · ∪ ∂O k , and we have (1.7). Also Applying (1.7) leads to two cases: In the first case, (1.5) holds. Furthermore, In the second case, since ∂O 1 ∪ · · · ∪ ∂O k is compact and (1.2) holds, we have ∂O 1 ∪ · · · ∪ ∂O k = J . (1.12) Going further, let us note that R and R k have the same Julia set and the same maximal entropy measure. Given (1.3), we have R k : O j → O j for each j ∈ Z/(k), hence R k : ∂O j → ∂O j . Hence, by Lemma 1.2 (applied to R k ), for each such j, λ(∂O j ) = 0 or 1. (1.13) (1.14) We still have (1.9), so This proves Theorem 1.1.
The following result of [3] is a significant consequence of Theorem 1. As noted in [3], it follows from Theorem 1.3 that and we have the following basic result of [9].
To establish the results in the last sentence of Corollary 1.4, we note that denseness of J 0 in J follows from (1.20) and (1.2). The fact that J 0 is obtained from J by successively removing ∂O j , j ∈ N, implies J 0 is a G δ subset of J .
Our goal is the study of λ(∂O) for various Fatou components in cases where this measure is not identically zero on these boundaries. An important class of rational maps with empty residual Julia set is the class of polynomials of degree d ≥ 2, which we tackle in §2. In this case, the Fatou component O ∞ containing ∞ satisfies We demonstrate this for a class of polynomials whose Fatou sets have an infinite number of components. One important property of O ∞ used in the analysis is its complete invariance: In §3 we extend the scope of this work to include other rational maps R for which there is a completely invariant Fatou component O # , so (1.23) We extend the basic results of §2 to cover this more general situation. In §4 we consider hyperbolic maps and geometrically finite maps. Results of § §3-4 together yield the following. The special case when R is a polynomial and J is connected and locally connected is Proposition 2.3. As we show in §4, using (B1), we can also replace (B) by J is connected and R is geometrically finite.
We end with an appendix, giving examples to illustrate our results.

Polynomials
Here we explore consequences of Theorem 1.1 for the class of polynomials of degree d ≥ 2, e.g., It clearly follows from (2.2) that for polynomials, Of course, Proposition 2.1 applies if K is not connected. We will now assume K is connected. Hence C \ K is simply connected. By the Riemann mapping theorem, there is a unique biholomorphic map Here D is the disk {z ∈ C : |z| ≤ 1}. We use this to recall from Chapter 6 of [7] conditions under which the hypothesis of Proposition 2.1 holds. It involves the notion of external rays, We mention two classical results, given as Theorems 6.1 and 6.2 of [7]. The first is that the limit in (2.6) exists for almost all θ ∈ T 1 . The second is the following: (2.7) These results lead to the following, which is part of Corollary 6.7 of [7]. (2.8) Proof. The union γ of the two rays {ψ ξ 0 (r) : 1 < r < ∞} and {ψ ξ 1 (r) : 1 < r < ∞}, together with their endpoints z 0 and ∞, forms a simple closed curve in C. The Jordan curve theorem implies that C \ γ has exactly two connected components. Our hypotheses imply It remains to note that each connected component of C \ γ contains a point of K, and this follows from (2.7), first taking E to be the open arc from ξ 0 to ξ 1 in T 1 , then taking E to be the complementary open arc in T 1 .
Regarding the question of when Proposition 2.2 applies, we note that a definite answer can be given under the additional hypothesis that J is connected and locally connected, (2.10) or equivalently, that K is connected and locally connected. In that case, a classical result of Caratheodory (cf. [8], Theorem 17.14) implies that ϕ in ( Now if ϕ in (2.12) is also one-to-one, this makes J ⊂ C a simple closed curve, so F = C \ J would have just two connected components. We have the following conclusion. Proof. As we have just seen, the hypotheses imply that ϕ in (2.12) is not one-to-one. Hence there exist ξ 0 = ξ 1 ∈ T 1 such that ϕ(e iξ 0 ) = ϕ(e iξ 1 ) in (2.13). Thus, with z 0 = ϕ(e iξ 0 ) = ϕ(e iξ 1 ), Proposition 3.2 implies that K \ {z 0 } is not connected, so (2.14) follows from Proposition 2.1.

Other maps with a completely invariant Fatou component
Extending our scope a bit, let us now assume that R (of degree d ≥ 2) has the property that F has a completely invariant connected component O # , i.e., If there is a fixed point p ∈ O # of R, then, conjugating by a linear fractional transformation, we can take p = ∞. If R −1 (p) = p as well, then R is a polynomial (which guarantees that p must be a superattracting fixed point  In counterpoint, we have the following result when J is not connected. Remark. As seen in §1, the hypothesis that λ(∂O 1 ) = 0 for some Fatou component O 1 is equivalent to the hypothesis that J 0 = ∅. The Makienko conjecture can be stated as saying that, if J 0 = ∅, then there is a Fatou component that is completely invariant under R 2 . A discussion of conditions under which Makienko's conjecture has been proved can be found in [2]. Of particular use here is the following result of [11]: The Makienko conjecture holds provided J is locally connected. (3.3) We make use of this in the following section.

Hyperbolic maps and geometrically finite maps
A rational map R : C → C, of degree d ≥ 2, with Julia set J , is said to be hyperbolic provided  Incidentally, this forces O 1 = O # . Furthermore, taking into account (1.6), we see that k must be 1. Hence O 1 must be completely invariant under R.
For further results, we bring in the following two generalizations of hyperbolicity, taking definitions from [10], p. 153.
Definition. A rational map R : C → C of degree ≥ 2 is said to be subhyperbolic provided the following two conditions hold: the forward orbit of each critical point in F converges to an attracting cycle, (4.4) and the forward orbit of each critical point in J is eventually periodic. (4.5) If we merely assume (4.5) holds, we say R is geometrically finite.
Clearly R hyperbolic ⇒ R subhyperbolic ⇒ R geometrically finite. The usefulness of geometric finiteness for the work here stems from the following generalization of (4.2), established in [12], Theorem A: If R is geometrically finite and J is connected, then J is locally connected. (4.6) In concert with (3.3), this yields the following:

A Examples
We include a few basic examples illustrating the results of § §1-4. In all these examples, J 0 = ∅. Before getting to them, we make a few useful general comments whose proofs are found in the literature in the bibliography. If R is a rational map of degree d ≥ 2 with a k-periodic point z 0 whose multiplier satisfies: DR k (z 0 ) = exp(2πi/m), m ∈ N, then the k-cycle is called parabolic; k always refers to the minimum period.
Remarks A.
1. If a rational map R has an attracting k-periodic point in F, then there are at least k Fatou components in F.

2.
If R has a parabolic k-periodic point z 0 with multiplier DR k (z 0 ) = exp(2πi/m), then there are at least mk Fatou components in F.
3. These lead to the following lemma.
Lemma A.1 Let R : C → C be a rational map of degree ≥ 2. Assume its Fatou set F has a completely invariant component and that R has an attracting or parabolic k-periodic point with k ≥ 2. Then F has infinitely many components.
Proof. The completely invariant component cannot be one of the components in the n-cycle, n ≥ 2, of components in F induced by the non-repelling k-cycle for R so F has more than two components.
4. Whenever R : C → C is a rational map of degree 2, the Julia set J is either connected or totally disconnected.
5. If every critical point in J has a finite forward orbit (i.e., R is geometrically finite), then all non-repelling periodic points of R are attracting or parabolic. If R has degree d ≥ 2 and there are 2d − 2 critical points in F, then R is geometrically finite.

Example 1.
As an example of a non-polynomial map with a completely invariant Fatou component, neither hyperbolic nor subhyperbolic, but geometrically finite, we set: R has critical points c 1 = 1 and c 2 = −1.
For these maps ∞ is a fixed point with multiplier 1 so the map is parabolic; this implies that ∞, 0 ∈ J and there is an attracting petal in F which sits in a component O # ⊂ F containing a critical point. We have: Since c 1 ∈ O # and R(O # ) = O # , O # is a degree 2 branched cover of itself, so O # is completely invariant. The map R is geometrically finite, and J is connected so it is locally connected using (4.6). Then λ(∂O # ) = 1 and λ(∂O 1 ) = λ(∂O 2 ) = 0 by Proposition 3.2.
Because of the parabolic point at ∞, R is neither hyperbolic nor subhyperbolic, since lim n→∞ R n (c 1 ) ∈ J causing (4.1) and (4.4) to fail.
Example 4. There are many cubic polynomials to which Theorem 1.5 (A) applies. In particular, there are cubic polynomials with the property that one critical point is attracted to ∞ and the other stays bounded. For these we frequently see that the Julia set is disconnected but not Cantor. Then Proposition 3.3 applies in this case.