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Dimensions of stable sets and scrambled sets in positive finite entropy systems

Published online by Cambridge University Press:  28 April 2011

CHUN FANG ,
WEN HUANG ,
YINGFEI YI  and
PENGFEI ZHANG
CHUN FANG
Affiliation:
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, PR China (email: Fangchun@mail.ustc.edu.cn, wenh@mail.ustc.edu.cn, pfzhang5@mail.ustc.edu.cn)
WEN HUANG
Affiliation:
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, PR China (email: Fangchun@mail.ustc.edu.cn, wenh@mail.ustc.edu.cn, pfzhang5@mail.ustc.edu.cn)
YINGFEI YI
Affiliation:
School of Mathematics, Jilin University, Changchun, 130012, PR China (email: yi@math.gatech.edu) School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA
PENGFEI ZHANG
Affiliation:
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, PR China (email: Fangchun@mail.ustc.edu.cn, wenh@mail.ustc.edu.cn, pfzhang5@mail.ustc.edu.cn)
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Abstract

We study the dimensions of stable sets and scrambled sets of a dynamical system with positive finite entropy. We show that there is a measure-theoretically ‘large’ set containing points whose sets of ‘hyperbolic points’ (i.e. points lying in the intersections of the closures of the stable and unstable sets) admit positive Bowen dimension entropies; under the continuum hypothesis, this set also contains a scrambled set with positive Bowen dimension entropies. For several kinds of specific invertible dynamical systems, the lower bounds of the Hausdorff dimension of these sets are estimated. In particular, for a diffeomorphism on a smooth Riemannian manifold with positive entropy, such a lower bound is given in terms of the metric entropy and Lyapunov exponent.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 2: Daniel J. Rudolph – in Memoriam , April 2012 , pp. 599 - 628
Copyright
Copyright © Cambridge University Press 2011

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