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Nonadditive measure-theoretic pressure and applications to dimensions of an ergodic measure

Published online by Cambridge University Press:  17 April 2012

YONGLUO CAO ,
HUYI HU  and
YUN ZHAO
YONGLUO CAO
Affiliation:
Department of Mathematics, Soochow University, Suzhou 215006, Jiangsu, PR China (email: ylcao@suda.edu.cn, zhaoyun@suda.edu.cn)
HUYI HU
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA (email: hu@math.msu.edu)
YUN ZHAO
Affiliation:
Department of Mathematics, Soochow University, Suzhou 215006, Jiangsu, PR China (email: ylcao@suda.edu.cn, zhaoyun@suda.edu.cn)
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Abstract

Without any additional conditions on subadditive potentials, this paper defines subadditive measure-theoretic pressure, and shows that the subadditive measure-theoretic pressure for ergodic measures can be described in terms of measure-theoretic entropy and a constant associated with the ergodic measure. Based on the definition of topological pressure on non-compact sets, we give another equivalent definition of subadditive measure-theoretic pressure, and obtain an inverse variational principle. This paper also studies the superadditive measure-theoretic pressure which has similar formalism to the subadditive measure-theoretic pressure. As an application of the main results, we prove that an average conformal repeller admits an ergodic measure of maximal Hausdorff dimension. Furthermore, for each ergodic measure supported on an average conformal repeller, we construct a set whose dimension is equal to the dimension of the measure.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 3 , June 2013 , pp. 831 - 850
Copyright
Copyright © 2012 Cambridge University Press

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