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Measures of maximal and full dimension for smooth maps

Part of: Smooth dynamical systems: general theory Infinite-dimensional dissipative dynamical systems Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  17 March 2023

YURONG CHEN ,
CHIYI LUO  and
YUN ZHAO
YURONG CHEN
Affiliation:
School of Mathematical Sciences, Soochow University, Suzhou 215006, Jiangsu, P. R. China and Center for Dynamical Systems and Differential Equations, Soochow University, Suzhou 215006, Jiangsu, P. R. China (e-mail: 20184207019@stu.suda.edu.cn, luochiyi98@gmail.com)
CHIYI LUO
Affiliation:
School of Mathematical Sciences, Soochow University, Suzhou 215006, Jiangsu, P. R. China and Center for Dynamical Systems and Differential Equations, Soochow University, Suzhou 215006, Jiangsu, P. R. China (e-mail: 20184207019@stu.suda.edu.cn, luochiyi98@gmail.com)
YUN ZHAO*
Affiliation:
School of Mathematical Sciences, Soochow University, Suzhou 215006, Jiangsu, P. R. China and Center for Dynamical Systems and Differential Equations, Soochow University, Suzhou 215006, Jiangsu, P. R. China (e-mail: 20184207019@stu.suda.edu.cn, luochiyi98@gmail.com)
*
(e-mail: zhaoyun@suda.edu.cn)
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Abstract

For a $C^1$ non-conformal repeller, this paper proves that there exists an ergodic measure of full Carathéodory singular dimension. For an average conformal hyperbolic set of a $C^1$ diffeomorphism, this paper constructs a Borel probability measure (with support strictly inside the repeller) of full Hausdorff dimension. If the average conformal hyperbolic set is of a $C^{1+\alpha }$ diffeomorphism, this paper shows that there exists an ergodic measure of maximal dimension.

Keywords

measure of full dimensionmeasure of maximal dimensionrepellerhyperbolic sets

MSC classification

Primary: 37C45: Dimension theory of dynamical systems 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
Secondary: 37L30: Attractors and their dimensions, Lyapunov exponents

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 44 , Issue 1 , January 2024 , pp. 31 - 49
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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