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Dimension approximation in smooth dynamical systems

Part of: Dynamical systems with hyperbolic behavior Smooth dynamical systems: general theory

Published online by Cambridge University Press:  11 April 2023

YONGLUO CAO ,
JUAN WANG  and
YUN ZHAO
YONGLUO CAO
Affiliation:
Departament of Mathematics, Soochow University, Suzhou 215006, Jiangsu, PR China (e-mail: ylcao@suda.edu.cn) Center for Dynamical Systems and Differential Equations, Soochow University, Suzhou 215006, Jiangsu, PR China
JUAN WANG
Affiliation:
School of Mathematics, Physics and Statistics, Shanghai University of Engineering Science, Shanghai 201620, PR China (e-mail: wangjuanmath@sues.edu.cn)
YUN ZHAO*
Affiliation:
Departament of Mathematics, Soochow University, Suzhou 215006, Jiangsu, PR China (e-mail: ylcao@suda.edu.cn) Center for Dynamical Systems and Differential Equations, Soochow University, Suzhou 215006, Jiangsu, PR China
*
e-mail: zhaoyun@suda.edu.cn
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Abstract

For a non-conformal repeller $\Lambda $ of a $C^{1+\alpha }$ map f preserving an ergodic measure $\mu $ of positive entropy, this paper shows that the Lyapunov dimension of $\mu $ can be approximated gradually by the Carathéodory singular dimension of a sequence of horseshoes. For a $C^{1+\alpha }$ diffeomorphism f preserving a hyperbolic ergodic measure $\mu $ of positive entropy, if $(f, \mu )$ has only two Lyapunov exponents $\unicode{x3bb} _u(\mu )>0>\unicode{x3bb} _s(\mu )$, then the Hausdorff or lower box or upper box dimension of $\mu $ can be approximated by the corresponding dimension of the horseshoes $\{\Lambda _n\}$. The same statement holds true if f is a $C^1$ diffeomorphism with a dominated Oseledet’s splitting with respect to $\mu $.

Keywords

dimensionhyperbolic measurehorseshoerepeller

MSC classification

Primary: 37C45: Dimension theory of dynamical systems 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
Secondary: 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 44 , Issue 2 , February 2024 , pp. 383 - 407
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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