Quasisymmetries of Sierpiński carpet Julia sets

Abstract

We prove that if ξ is a quasisymmetric homeomorphism between Sierpiński carpets that are the Julia sets of postcritically-finite rational maps, then ξ is the restriction of a Möbius transformation to the Julia set. This implies that the group of quasisymmetric homeomorphisms of a Sierpiński carpet Julia set of a postcritically-finite rational map is finite.Theorem 1.10. Let f : C → C be a subhyperbolic rational map whose Julia set J (f ) is a Sierpiński carpet. Then the peripheral circles of the Sierpiński carpet J (f ) are uniform quasicircles, they are uniformly relatively separated, and they occur on all locations and scales. Moreover, J (f ) is a porous set, and in particular, has measure zero.See Sections 2 and 5 for an explanation of the terminology and for the proof.

On the dynamics of a family of singularly perturbed rational maps

“…In particular, the Julia set J(f λ ) is homeomorphic to Σ 3 × S 1 , which is a Cantor set of circles, as desired. This ends the proof of Theorem 4.4, Theorem 1.1 (2) and Theorem 1.2 in the case when J(f λ ) is a Cantor set of circles. 2…”

“…There are some important conjectures about the boundaries of hyperbolic groups and the Sierpiński carpets [10]. Recently, Bonk, Lyubich and Merenkov studied the quasisymmetric geometry on the carpet Julia sets of postcritically-finite rational maps [2]. In this section, we will give a sufficient and necessary condition when the Julia set of f λ is a Sierpiński carpet by studying the "escaping times" of the free critical points.…”

Square Sierpiński carpets and Lattès maps

“…Let A be the set of all points in J (g) that lie on a peripheral circle of J (g). If J is a peripheral circle of J (g), then g(J) is also a peripheral circle and g −1 (J) consists of finitely many peripheral circles (see [BLM,Lemma 5.1]). This implies that A is completely invariant under g, and hence under all iterates of g, i.e., g l (A) = A = g −l (A) for each l ∈ N. Note also that the homeomorphism ξ sends the peripheral circles of D p to the peripheral circles of J (g).…”

“…In order to establish (7.4), one uses ideas as in the proof of Proposition 5.1 combined with arguments for the proof of the similar relation (8.4) in [BLM,Section 8], where T plays the role of the rational map f and D p the role of the Julia set J (f ).…”

A criterion to generate carpet Julia sets

“…On the other hand, in the geometry group theory, the quasisymmetric equivalences of the Sierpiński carpets has been studied extensively since it was partially motivated by the Kapovich-Kleiner conjecture [KK00]. Recently, the quasisymmetric geometry of the carpet Julia sets has also been considered in [BLM16], [QYZ14] and [QYY16].…”