Quasisymmetric geometry of the Cantor circles as the Julia sets of rational maps

Weiyuan  Qiu; Fei  Yang; Yongcheng  Yin

doi:10.3934/dcds.2016.36.3375

Article Contents

2016, Volume 36, Issue 6: 3375-3416. Doi: 10.3934/dcds.2016.36.3375

This issue Previous Article On elliptic systems with Sobolev critical exponent Next Article Hyperbolic sets and entropy at the homological level

Quasisymmetric geometry of the Cantor circles as the Julia sets of rational maps

1.
School of Mathematical Sciences, Fudan University, Shanghai 200433
2.
Department of Mathematics, Nanjing University, Nanjing 210093
3.
Department of Mathematics, Zhejiang University, Hangzhou 310027

Received: April 2015

Revised: October 2015

Published: May 2017

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Abstract

We give three families of parabolic rational maps and show that every Cantor set of circles as the Julia set of a non-hyperbolic rational map must be quasisymmetrically equivalent to the Julia set of one map in these families for suitable parameters. Combining a result obtained before, we give a complete classification of the Cantor circles Julia sets in the sense of quasisymmetric equivalence. Moreover, we study the regularity of the components of the Cantor circles Julia sets and establish a sufficient and necessary condition when a component of a Cantor circles Julia set is a quasicircle.

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Mathematics Subject Classification: Primary: 37F45; Secondary: 37F20, 37F10.

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