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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.

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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References

[1]Ban, J., Cao, Y. and Hu, H.. The dimensions of a non-conformal repeller and an average conformal repeller. Trans. Amer. Math. Soc. 362 (2010), 727–751.Google Scholar
[2]Barreira, L.. A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems. Ergod. Th. & Dynam. Sys. 16 (1996), 871–927.Google Scholar
[3]Barreira, L. and Wolf, C.. Measures of maximal dimension for hyperbolic diffeomorphisms. Comm. Math. Phys. 239 (2003), 93–113.Google Scholar
[4]Bowen, R.. Topological entropy for noncompact sets. Trans. Amer. Math. Soc. 184 (1973), 125–136.Google Scholar
[5]Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in Mathematics, 470). Springer, New York-Heidelberg-Berlin, 1975.Google Scholar
[6]Bowen, R.. Hausdorff dimension of quasicircles. Publ. Math. Inst. Hautes Études Sci. 50 (1979), 11–25.CrossRefGoogle Scholar
[7]Brin, M. and Katok, A.. On Local Entropy (Lecture Notes in Mathematics, 1007). Springer, New York, 1983.Google Scholar
[8]Cao, Y.. Nonzero Lyapunov exponents and uniform hyperbolicity. Nonlinearity 16 (2003), 1473–1479.Google Scholar
[9]Cao, Y.. Dimension estimate and multifractal analysis of $C^1$ average conformal repeller. Preprint.Google Scholar
[10]Cao, Y., Feng, D. and Huang, W.. The thermodynamic formalism for sub-multiplicative potentials. Discrete Contin. Dyn. Syst. Ser. A 20 (2008), 639–657.Google Scholar
[11]Cheng, W., Zhao, Y. and Cao, Y.. Pressures for asymptotically subadditive potentials under a mistake function. Discrete Contin. Dyn. Syst. Ser. A 32 (2012), 487–497.Google Scholar
[12]Climenhaga, V.. Bowen’s equation in the non-uniform setting. Ergod. Th. & Dynam. Sys. 31 (2011), 1163–1182.Google Scholar
[13]Falconer, K.. A subadditive thermodynamic formalism for mixing repellers. J. Phys. A 21 (1988), 737–742.Google Scholar
[14]Gatzouras, D. and Peres, Y.. The Variational Principle for Hausdorff Dimension: a Survey (London Mathematical Society Lecture Note Series, 228). Cambridge University Press, Cambridge, 1996, pp. 113–126.Google Scholar
[15]He, L., Lv, J. and Zhou, L.. Definition of measure-theoretic pressure using spanning sets. Acta Math. Sin. (Engl. Ser.) 20 (2004), 709–718.Google Scholar
[16]Huang, W. and Zhang, P.. Pointwise dimension, entropy and Lyapunov exponents for $C^1$ map. Trans. Amer. Math. Soc. to appear.Google Scholar
[17]Katok, A.. Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Inst. Hautes Études Sci. Publ. Math. 51 (1980), 137–173.Google Scholar
[18]Kingman, J. F. C.. The ergodic theory of subadditive stochastic processes. J. R. Stat. Soc. Ser. B 30 (1968), 499–510.Google Scholar
[19]Ledrappier, F.. Some relations between dimension and Lyapunov exponent. Comm. Math. Phys. 81 (1981), 229–238.Google Scholar
[20]Luzia, N.. A variational principle for the dimension for a class of non-conformal repellers. Ergod. Th. & Dynam. Sys. 26 (2006), 821–845.Google Scholar
[21]Oseledec, V.. A Multiplicative Ergodic Theorem, Characteristic Lyapnov Exponents of Dynamical Systems (Transactions of the Moscow Mathematical Society, 19). American Mathematical Society, Providence, RI, 1968.Google Scholar
[22]Pesin, Ya.. Dimension Theory in Dynamical Systems, Contemporary Views and Applications. University of Chicago Press, Chicago, 1997.Google Scholar
[23]Pesin, Ya. and Pitskel’, B.. Topological pressure and the variational principle for noncompact sets. Funct. Anal. Appl. 18 (1984), 307–318.Google Scholar
[24]Ruelle, D.. Statistical mechanics on a compact set with $Z^{\nu }$ action satisfying expansiveness and specification. Trans. Amer. Math. Soc. 187 (1973), 237–251.Google Scholar
[25]Ruelle, D.. Repellers for real analytic maps. Ergod. Th. & Dynam. Sys. 2 (1982), 99–107.Google Scholar
[26]Walters, P.. A variational principle for the pressure of continuous transformations. Amer. J. Math. 97 (1975), 937–971.Google Scholar
[27]Walters, P.. An Introduction to Ergodic Theory. Springer, New York, 1982.Google Scholar
[28]Zhang, G.. Variational principles of pressure. Discrete Contin. Dyn. Syst. Ser. A 24 (2009), 1409–1435.Google Scholar
[29]Zhang, Y.. Dynamical upper bounds for Hausdorff dimension of invariant sets. Ergod. Th. & Dynam. Sys. 17 (1997), 739–756.Google Scholar
[30]Zhao, Y.. A note on the measure-theoretic pressure in subadditive case. Chin. Ann. Math. Ser. A (3) 29 (2008), 325–332.Google Scholar
[31]Zhao, Y. and Cao, Y.. On the topological pressure of random bundle transformations in subadditive case. J. Math. Anal. Appl. 342 (2008), 715–725.Google Scholar
[32]Zhao, Y. and Cao, Y.. Measure-theoretic pressure for subadditive potentials. Nonlinear Anal. 70 (2009), 2237–2247.Google Scholar
[33]Zhao, Y., Cao, Y. and Ban, J.. The Hausdorff dimension of average conformal repellers under random perturbation. Nonlinearity 22 (2009), 2405–2416.Google Scholar