Hölder exponents of the Green functions of planar polynomial Julia sets

If the Green function gE of a compact set E ⊂ C is Hölder continuous, then the Hölder exponent of the set E is the supremum over all such α that |gE (z) − gE (w)| ≤ M |z − w|α, z, w ∈ C. We give a lower bound for the Hölder exponent of the Julia sets of polynomials. In particular, we show that there exist totally disconnected planar sets with the Hölder exponent greater than 1/2 as well as fat continua with the boundary nowhere smooth and with the Hölder exponent as close to 1 as we wish.


Introduction
The continuity of the Green function of a planar compact set (which is actually the Green function of the bounded component of the complement of the compact set to the Riemann sphere) was always the object of an intensive research. In what follows, we investigate only compact planar sets with Hölder continuous Green functions, that is, we take such a compact set E ⊂ C that there exist positive constants M and α with (1.1) In this situation, there arises another problem: to find estimates for the exponent α. To this end, we define the Hölder exponent of the set E to be λ(E) := sup{α : (1.1) holds with the exponent α} and look for its estimates.
In this note, we deal with polynomial Julia sets, that is, the Julia sets associated with polynomials of degree d ≥ 2. They are compact and regular. The proof of the Hölder continuity of their Green functions was provided by Sibony (see [6, Chapter VIII, Theorem 3.2] and the comments near it).
The main result of this paper runs as follows .
The most known and investigated are Julia sets J c ⊂ C and their filled-in counterparts K c defined by the quadratic polynomials of the form z → z 2 + c, c ∈ C. We will obtain Note that this estimate is sharp in the following sense. Namely λ(K 0 ) ≥ 1 and λ(K −2 ) ≥ 1/2 in view of Corollary 1.2. Since K 0 = {z ∈ C : |z| ≤ 1}, it is known that λ(K 0 ) = 1. On the other hand, K −2 = [−2, 2], and hence it is known that λ(K −2 ) = 1/2. The estimate given in Theorem 1.2 is especially worthwhile when K c is totally disconnected or if |c| is small (small enough to give the estimate greater than 1/2).
What is already known about the Hölder exponent λ(E) of a compact set E ⊂ C (with the Hölder continuous Green function)? We have always (see [16,Remark 3.7]) Furthermore, we have also (see [15,Theorem 2]) where dim H (E) denotes the Hausdorff dimension of the set E.
On the other hand, if E ⊂ C is a non-degenerate continuum, then (by a continuum, we mean a connected compact set). This inequality remains true if E ⊂ C is a compact set satisfying the following condition: there exists a positive constant δ such that each point a ∈ E belongs to a subset F of E, which is a continuum of the diameter greater than δ. This result follows from Leja Polynomial Lemma (cf. [11]). Finally, let us mention one of the most important applications of the Hölder exponent in the theory of polynomial inequalities. Namely, if the Green function g E is Hölder continuous, then E is a Markov set (for the background see e.g. [13]), that is, there exist positive constants M, m such that for all polynomials p (1.5) For such sets, the Markov exponent, m(E) := inf{m > 0 : (1.5) holds with the exponent m}, was defined in [2] (for its importance, good references and some new results see [1]). The following relation is true: .
By a fat set, we mean a compact set that coincides with the closure of its interior. Let us mention one important well-known fact.

Proposition 1.3 If E is a fat planar continuum with a piecewise analytic boundary, then
Proof It is known that m(E) = 1 ⇐⇒ λ(E) = 1 if E is a planar compact set with Hölder continuous Green function. And m(E) = 1 for any fat planar continuum E with a piecewise analytic boundary (see [17]).
Note that while obtaining bounds for the Hölder exponent, we gain also estimates for the Markov exponent by (1.6) and for the Hausdorff dimension of the set by (1.3). This paper was stimulated by Baran, who asked whether it is possible to find the estimates for the filled-in Julia sets, and also by the paper [15], where some estimates for the Hölder exponent of the Cantor ternary sets were given. The Cantor ternary sets can be viewed (and actually the authors of [15] use this fact) as attractors of iterated function systems. Julia sets can actually be obtained in a similar way even if the methods for getting the estimates are different.

Preliminaries
Put D(a, r ) := {z ∈ C : |z − a| ≤ r } for a ∈ C and r > 0. For a compact set F ⊂ C and a positive number r > 0 define The Green function g E of the compact set E ⊂ C (of positive logarithmic capacity) can be defined in the same way as the function V E in [10, Chapter 5] or as the Green's function of the unbounded component of C ∞ \ E with pole at ∞ and extended to be zero elsewhere on C (see e.g. [14,Chapter 4.4]). If h : C z → az + b ∈ C for some complex number b and non-zero complex number a, then If a set E ∈ C is compact, its polynomially convex hull is denoted by E. We have g E ≡ g E by definition (since both can be defined as the Green function of the complement of E to C ∞ ). Hence, λ(E) = λ( E). Moreover, by the harmonicity of the Green function in order to prove the Hölder continuity (with the exponent α > 0) of the Green function of a compact set E, it suffices to show that there exist constants , M > 0 such that the following inequality holds Chapter VIII, the proof of Theorem 3.2]). Inequality (2.1) is called the Hölder Continuity Property.
Let p : C → C be a complex polynomial of degree d ≥ 2. The Julia set J [ p] is usually defined in the terms of (non-)normality of the family { p n } (see e.g. [3] or [6]). We will use another equivalent way. We define first the filled-in Julia set associated with p to be ). The following transformation formula can be obtained, for example, from [10, Theorem 5.3.1].

Proof of the main result
Proof (of Main Theorem 1.1) Fix ε > 0 and define Note that for any positive integer n.
Since A(ε) α(ε) = d, we derive from (2.2) by induction that In view of (3.1) by the mean value property and the definition of M(ε) yields Note that in the last inequality, there is no dependence on m = m(z).
Remark 3.1 Note that it does not follow from the proof of Main Theorem 1.1 that g K [ p] is Hölder continuous with the exponent α, since if ε −→ 0, then M(ε) tends to infinity.

Quadratic polynomials
We start with the announced filled-in Julia sets for quadratic polynomials. For the background see, for example, [6]. It is well known that it suffices to consider the polynomials of the form Q c : z → z 2 + c, c ∈ C, since any other quadratic polynomial is conjugated to one of this type. Let J c denote the Julia set of Q c and K c -the filled-in one. We will now prove Corollary 1.2.  We shall start with the mentioned special cases  Proof It follows from Corollary 1.2 that λ(K c ) ≥ log 2 log 4 = 1 2 . By the famous Markov inequality, it is known that m([−2, 2]) = 2, hence λ(K −2 ) = 1/2 by (1.6).

Remark 4.4
Note that K −2 is connected but if |c| = 2 and c = −2, then K c is totally disconnected. However, its Hölder exponent satisfies the same inequality as all those of connected sets.
Before we list the estimates for some other values of c, let us recall some facts. The Mandelbrot set can be defined in two ways We have D(0, 1 4 ) ⊂ M, thus in particular if |c| ≤ 1/4, then the filled-in Julia set K c is connected. What is more if c is not real and lies in the interior of the main cardioid C 1 (see [8,Fig.17.4]), the Julia set J c is a simple closed curve that contains no smooth (i.e., of class C 1 ) arcs (see [7, Proposition 3.6.3 and the Remark after it]). Therefore, if we get the bound greater than 1/2 of the Hölder exponent, then it is also interesting for such c.

then J c is a simple closed curve which contains no smooth arcs and λ(J c ) > 2/3.
Proof This is the straightforward consequence of Corollary 1.2 and the facts above, since |c| < 1/4 yields c ∈ C 1 .
The following result is especially noteworthy in comparison with Proposition 1.3.

Corollary 4.6
There exist fat continua with the boundaries that contain no smooth arcs and with the Hölder exponents as close to 1 as we wish.
Proof The lower bound in Corollary 1.2 depends continuously on |c| and tends to 1 when |c| → 0. Therefore, it suffices to take, for example, K it with t ∈ (0, 1/4) small enough.
The estimate given here is also remarkable for all c outside the Mandelbrot set, since then J c = K c is totally disconnected and we cannot use the bound (1.4).
Let us see first one example Example 4.7 If |c| = 15 4 , then λ(K c ) ≥ log 2 log 5 > 0, 43 and K c is totally disconnected. But this bound is not that impressive, since it is smaller than 1/2. Recall now that if c > 1/4, then J c = K c is totally disconnected and has Lebesgue measure 0 [5, Theorem 12.1].
One can also visit, for example, [9] for pictures of Julia sets for polynomials Q c , c ∈ C.

Cubic polynomials
Every cubic polynomial can be conjugated to one of the form thus it suffices to consider cubic polynomials of this type. Denote by J a,b and K a,b the Julia set and the filled-in Julia set of C a,b .
We will improve this result in some special cases.
Some pictures of cubic Julia sets can be find in [12]. Let us finally count some other examples ] has infinitely many non-degenerate components and λ(J [z → z 2 − z 3 /9]) > 3/8.

Proof
The first statement is from [3, Section 11.4]. The polynomial z → z 2 − z 3 /9 is conjugated to C 3,−i , so it suffices to apply Corollary 5.1.

Other polynomials
Let us note first Let us also recall that ∀n ∈ N : J [ p] = J [ p n ] ([7, Proposition 3.5.4]), thus in the previous sections, we already considered the Julia sets of some polynomials of higher degrees too. We propose some more examples whose proofs we omit. They are more or less based on the same argument for a quadratic function.