Quasisymmetrically inequivalent hyperbolic Julia sets

Abstract

Abstract. We give explicit examples of pairs of Julia sets of hyperbolic rational maps which are homeomorphic but not quasisymmetrically homeomorphic.Quasiconformal geometry is concerned with properties of metric spaces that are preserved under quasisymmetric homeomorphisms. Recall that a homeomorphism h : X → Y between metric spaces is quasisymmetric if there exists a distortion control function η : [0, ∞) → [0, ∞) which is a homeomorphism and which satisfies |h(x) − h(a)|/|h(x) − h(b)| ≤ η(|x − a|/|x − b|) for every triple of distinct points x, a, b ∈ X. We shall say that X and Y are quasisymmetrically equivalent if there exists such a homeomorphism.A basic -even if still widely open-question is to determine whether two given spaces belong to the same quasisymmetry class, once it is known that they are homeomorphic and share the same qualitative geometric properties. This question arises also in the classification of hyperbolic spaces and word hyperbolic groups in the sense of Gromov [BP, Kle, Haï]. Besides spaces modelled on manifolds, very few examples are understood; see nonetheless [Bou] for examples of inequivalent spaces modelled on the universal Menger curve. Here, we focus our attention on compact metric spaces that arise as Julia sets of rational maps. A rational map is hyperbolic if the closure of the set of forward orbits of all its critical points does not meet its Julia set. We address the question of whether the geometry of the Julia set of a hyperbolic rational map is determined by its topology. More precisely, given two hyperbolic rational maps f and g with homeomorphic Julia sets J f and J g , does there exist a quasisymmetric homeomorphism h :Hyperbolic Julia sets serve our purposes for several reasons. First, it rules out elementary local obstructions. For instance, the Julia set of f (z) = z 2 is the Euclidean unit circle S 1 , while that of g(z) = z 2 + 1/4 is a Jordan curve with a cusp at the unique fixed-point, so they are not quasisymmetrically equivalent. Second, if f is hyperbolic, it is locally invertible near J f , and the inverse branches are uniformly Date: July 2, 2018.

Exponential growth of some iterated monodromy groups

“…We do not know if these hyperbolic maps have IMGs of exponential growth. However, the argument in , showing that has a Sierpiński carpet Julia set, is equally valid independently of whether is even or odd.…”

“…The rational maps we will consider now have originally been studied by Haïssinsky and Pilgrim in (with one difference that we address at the end of the section). We briefly review the construction, which is similar to the way the map from Section 3 was defined.…”

“…So, is a rational Thurston map with . Moreover, the Julia set of is a Sierpiński carpet, see .…”

“…Again the resulting rational map has a Sierpiński carpet as its Julia set, and is of exponential growth. The maps in are constructed in the same fashion, but there is even. This results in a hyperbolic rational map , meaning that every critical point of is contained in the Fatou set .…”

On the dynamics of a family of singularly perturbed rational maps

“…Therefore, a natural question is to find other types of rational maps whose Julia sets are Cantor circles but the dynamics on the Julia sets are not topologically conjugate to any McMullen map on their corresponding Julia sets. This question was solved in [9] by Haïssinsky and Pilgrim. They proved the existence of such rational maps by quasiconformal surgery. Later, the specific expressions of these rational maps were given in [17].…”