# Local rigidity of Schottky maps

@inproceedings{Merenkov2013LocalRO, title={Local rigidity of Schottky maps}, author={Sergei Merenkov}, year={2013} }

We introduce Schottky maps-conformal maps between relative Schottky sets, and study their local rigidity properties. This continues the investigations of relative Schottky sets initiated in [S. Merenkov, "Planar relative Schottky sets and quasisymmetric maps", Proc. London Math. Soc. (3) 104 (2012), 455-485]. Besides being of independent interest, the latter and current works provide key ingredients in the forthcoming proof of quasisymmetric rigidity of Sierpi\'nski carpet Julia sets of…

## 12 Citations

### Quasisymmetric geometry of Sierpiński carpet Julia sets

- MathematicsFundamenta Mathematicae
- 2019

In this paper, the main focus is on the Sierpinski carpet Julia sets of the rational maps with non-recurrent critical points. We study the uniform quasicircle property of the peripheral circles, the…

### Square Sierpi\'nski carpets and Latt\`es maps

- Mathematics
- 2018

We prove that every quasisymmetric homeomorphism of a standard square Sierpi\'nski carpet $S_p$, $p\ge 3$ odd, is an isometry. This strengthens and completes earlier work by the authors. We also show…

### Square Sierpiński carpets and Lattès maps

- Mathematics
- 2018

We prove that every quasisymmetric homeomorphism of a standard square Sierpiński carpet $$S_p$$ S p , $$p\ge 3$$ p ≥ 3 odd, is an isometry. This strengthens and completes earlier work by the authors…

### Potential Theory on Sierpiński Carpets

- MathematicsLecture Notes in Mathematics
- 2020

Author(s): Ntalampekos, Dimitrios | Advisor(s): Bonk, Mario | Abstract: This research is motivated by the study of the geometry of fractal sets and is focused on uniformization problems:…

### Rigidity theorems for circle domains

- MathematicsInventiones mathematicae
- 2019

A circle domain $$\Omega $$ Ω in the Riemann sphere is conformally rigid if every conformal map from $$\Omega $$ Ω onto another circle domain is the restriction of a Möbius transformation. We show…

### REMARKS ON QUASISYMMETRIC RIGIDITY OF SQUARE SIERPIŃSKI CARPETS

- MathematicsFractals
- 2018

Let [Formula: see text] be the standard Sierpiński carpet and [Formula: see text] the group of quasisymmetric maps of [Formula: see text] onto itself, where [Formula: see text] is odd. Mario Bonk and…

### The geometry of the Sierpinski carpets as the Julia sets of rational maps

- Mathematics
- 2014

Let $f$ be a rational map whose Julia set $J(f)$ is a Sierpi\'{n}ski carpet. We prove that $J(f)$ is quasisymmetrically equivalent to a round carpet if the $\omega$-limit sets of the critical points…

### Local rigidity for hyperbolic groups with Sierpiński carpet boundaries

- MathematicsCompositio Mathematica
- 2014

Abstract Let $G$ and $\tilde{G}$ be Kleinian groups whose limit sets $S$ and $\tilde{S}$, respectively, are homeomorphic to the standard Sierpiński carpet, and such that every complementary component…

### Square Sierpiński carpets and Lattès maps

- Materials ScienceMathematische Zeitschrift
- 2019

We prove that every quasisymmetric homeomorphism of a standard square Sierpiński carpet Sp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts}…

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