Dimension estimates in non-conformal setting

Juan Wang; Yongluo Cao; Yun Zhao

doi:10.3934/dcds.2014.34.3847

Article Contents

2014, Volume 34, Issue 9: 3847-3873. Doi: 10.3934/dcds.2014.34.3847

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Dimension estimates in non-conformal setting

1.
Department of mathematics, Soochow University, Suzhou 215006, Jiangsu
2.
Department of Mathematics, Soochow University, Suzhou 215006, Jiangsu

Received: June 30, 2013

Revised: November 30, 2013

Published: August 2014

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Abstract

Given an arbitrary subset of a non-conformal repeller of a $C^1$ map. Using directly the definitions of Hausdorff dimension and Box dimension and of pressure, this paper first proves that the zeros of the topological pressure on this set give its dimension estimates. And without using the estimate of pointwise dimension of an ergodic measure on a non-conformal repeller, it is showed that the zeros of non-additive measure-theoretic pressure give the lower and upper bound of dimension estimate for it. Some results from [22,41] are extended for $C^1$ maps or arbitrary subsets of a non-conformal repeller. And a remark on Rugh's result [34] is also given in this paper.

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Mathematics Subject Classification: Primary: 37C45, 28D20; Secondary: 37B25.

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