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Escape components of McMullen maps

Part of: Complex dynamical systems

Published online by Cambridge University Press:  28 November 2022

WEIYUAN QIU ,
PASCALE ROESCH  and
YUEYANG WANG
WEIYUAN QIU
Affiliation:
School of Mathematical Sciences, Fudan University, Shanghai, 200433, P. R. China (e-mail: [email protected])
PASCALE ROESCH
Affiliation:
IMT, Laboratoire Emile Picard, Université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse Cedex 9, France (e-mail: [email protected])
YUEYANG WANG*
Affiliation:
Department of Mathematics Zhejiang University, Hangzhou, 310027, P. R. China
*
e-mail: [email protected]
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Abstract

We consider the McMullen maps $f_{\unicode{x3bb} }(z)=z^{n}+\unicode{x3bb} z^{-n}$ with $\unicode{x3bb} \in \mathbb {C}^{*}$ and $n \geq 3$. We prove that the closures of escape hyperbolic components are pairwise disjoint and the boundaries of all bounded escape components (the McMullen domain and Sierpiński holes) are quasi-circles with Hausdorff dimension strictly between $1$ and $2$.

Keywords

hyperbolic componentMcMullen mapquasicircleholomorphic motion

MSC classification

Primary: 37F45: Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations
Secondary: 37F10: Polynomials; rational maps; entire and meromorphic functions 37F15: Expanding maps; hyperbolicity; structural stability
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 11 , November 2023 , pp. 3745 - 3775
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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Footnotes

*

The original version of this article contained an error in the name Pascale Roesch. This error has been corrected. A notice detailing this error has been published.

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