Dimensions of $ C^1- $average conformal hyperbolic sets

Juan Wang; Jing Wang; Yongluo Cao; Yun Zhao

doi:10.3934/dcds.2020065

Article Contents

2020, Volume 40, Issue 2: 883-905. Doi: 10.3934/dcds.2020065

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Dimensions of $ C^1- $average conformal hyperbolic sets

1.
School of Mathematics, Physics and Statistics, Shanghai University of Engineering Science, Shanghai 201620, China
2.
Departament of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200062, China
3.
Department of Mathematics, Soochow University, Suzhou 215006, Jiangsu, China
4.
Center for Dynamical Systems and Differential Equation, Soochow University, Suzhou 215006, Jiangsu, China

^* Corresponding author: Yun Zhao
^* Corresponding author: Yun Zhao

Received: March 2019

Revised: August 2019

Published: November 2019

The first author is supported by NSFC (11501400, 11871361) and the Talent Program of Shanghai University of Engineering Science. The third author is partially supported by NSFC (11771317, 11790274) and Science and Technology Commission of Shanghai Municipality (No. 18dz2271000). The fourth author is partially supported by NSFC (11871361, 11790274)

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Abstract

This paper introduces the concept of average conformal hyperbolic sets, which admit only one positive and one negative Lyapunov exponents for any ergodic measure. For an average conformal hyperbolic set of a $ C^1 $ diffeomorphism, utilizing the techniques in sub-additive thermodynamic formalism and some geometric arguments with unstable/stable manifolds, a formula of the Hausdorff dimension and lower (upper) box dimension is given in this paper, which is exactly the sum of the dimensions of the restriction of the hyperbolic set to stable and unstable manifolds. Furthermore, the dimensions of an average conformal hyperbolic set vary continuously with respect to the dynamics.

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Mathematics Subject Classification: Primary: 37C45, 37D20; Secondary: 37D35.

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