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Bowen’s equation in the non-uniform setting

Published online by Cambridge University Press:  20 July 2010

VAUGHN CLIMENHAGA
VAUGHN CLIMENHAGA*
Affiliation:
Department of Mathematics, McAllister Building, Pennsylvania State University, University Park, PA 16802, USA (email: climenha@math.psu.edu)
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Abstract

We show that Bowen’s equation, which characterizes the Hausdorff dimension of certain sets in terms of the topological pressure of an expanding conformal map, applies in greater generality than has been heretofore established. In particular, we consider an arbitrary subset Z of a compact metric space and require only that the lower Lyapunov exponents be positive on Z, together with a tempered contraction condition. Among other things, this allows us to compute the dimension spectrum for Lyapunov exponents for maps with parabolic periodic points, and to relate the Hausdorff dimension to the topological entropy for arbitrary subsets of symbolic space with the appropriate metric.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 4 , August 2011 , pp. 1163 - 1182
Copyright
Copyright © Cambridge University Press 2010

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References

[1]Barreira, L., Pesin, Y. and Schmeling, J.. On a general concept of multifractality: multifractal spectra for dimensions, entropies, and Lyapunov exponents. Multifractal rigidity. Chaos 7(1) (1997), 27–38.CrossRefGoogle ScholarPubMed
[2]Barreira, L. and Schmeling, J.. Sets of non-typical points have full topological entropy and full Hausdorff dimension. Israel J. Math. 116 (2000), 29–70.CrossRefGoogle Scholar
[3]Bowen, R.. Topological entropy for noncompact sets. Trans. Amer. Math. Soc. 184 (1973), 125–136.CrossRefGoogle Scholar
[4]Bowen, R.. Hausdorff dimension of quasicircles. Publ. Math. Inst. Hautes Études Sci. 50 (1979), 11–25.CrossRefGoogle Scholar
[5]Climenhaga, V.. Multifractal formalism derived from thermodynamics. Preprint, 2010.CrossRefGoogle Scholar
[6]Denker, M. and Urbański, M.. Ergodic theory of equilibrium states for rational maps. Nonlinearity 4 (1991), 103–134.CrossRefGoogle Scholar
[7]Gatzouras, D. and Peres, Y.. Invariant measures of full dimension for some expanding maps. Ergod. Th. & Dynam. Sys. 17(1) (1997), 147–167.CrossRefGoogle Scholar
[8]Gelfert, K., Przytycki, F. and Rams, M.. Lyapunov spectrum for rational maps. Preprint, 2009.CrossRefGoogle Scholar
[9]Gelfert, K. and Rams, M.. The Lyapunov spectrum of some parabolic systems. Ergod. Th. & Dynam. Sys. 29 (2009), 919–940.CrossRefGoogle Scholar
[10]Hu, H.. Equilibriums of some non-Hölder potentials. Trans. Amer. Math. Soc. 360(4) (2008), 2153–2190.CrossRefGoogle Scholar
[11]Iommi, G. and Kiwi, J.. The Lyapunov spectrum is not always concave. J. Stat. Phys. 135(3) (2009), 535–546.CrossRefGoogle Scholar
[12]Makarov, N. and Smirnov, S.. On thermodynamics of rational maps I. Negative spectrum. Comm. Math. Phys. 211 (2000), 705–743.CrossRefGoogle Scholar
[13]Mayer, V. and Urbański, M.. Geometric thermodynamic formalism and real analyticity for meromorphic functions of finite order. Ergod. Th. & Dynam. Sys. 28(3) (2008), 915–946.CrossRefGoogle Scholar
[14]Mayer, V. and Urbański, M.. Thermodynamical formalism and multifractal analysis for meromorphic functions of finite order. Mem. Amer. Math. Soc. 203(954) (2010), 1–107.Google Scholar
[15]Nakaishi, K.. Multifractal formalism for some parabolic maps. Ergod. Th. & Dynam. Sys. 20(3) (2000), 843–857.CrossRefGoogle Scholar
[16]Pesin, Ya. B. and Pitskel ′, B. S.. Topological pressure and the variational principle for noncompact sets. Funktsional. Anal. i Prilozhen. 18(4) (1984), 50–63 96.CrossRefGoogle Scholar
[17]Pesin, Y.. Dimension Theory in Dynamical Systems: Contemporary Views and Applications. University of Chicago Press, Chicago, 1998.Google Scholar
[18]Pesin, Y. and Weiss, H.. The multifractal analysis of Gibbs measures: motivation, mathematical foundation, and examples. Chaos 7(1) (1997), 89–106.CrossRefGoogle ScholarPubMed
[19]Pollicott, M. and Weiss, H.. Multifractal analysis of Lyapunov exponent for continued fraction and Manneville–Pomeau transformations and applications to Diophantine approximation. Comm. Math. Phys. 207(1) (1999), 145–171.CrossRefGoogle Scholar
[20]Przytycki, F., Rivera-Letelier, J. and Smirnov, S.. Equivalence and topological invariance of conditions for non-uniform hyperbolicity in the iteration of rational maps. Invent. Math. 151(1) (2003), 29–63.CrossRefGoogle Scholar
[21]Przytycki, F., Rivera-Letelier, J. and Smirnov, S.. Equality of pressures for rational functions. Ergod. Th. & Dynam. Sys. 24(3) (2004), 891–914.CrossRefGoogle Scholar
[22]Ruelle, D.. Repellers for real analytic maps. Ergod. Th. & Dynam. Sys. 2(1) (1982), 99–107.CrossRefGoogle Scholar
[23]Rugh, H. H.. On the dimensions of conformal repellers. Randomness and parameter dependency. Ann. of Math. (2) 168(3) (2008), 695–748.CrossRefGoogle Scholar
[24]Urbański, M.. On the Hausdorff dimension of a Julia set with a rationally indifferent periodic point. Studia Math. 97(3) (1991), 167–188.CrossRefGoogle Scholar
[25]Urbański, M.. Parabolic Cantor sets. Fund. Math. 151(3) (1996), 241–277.Google Scholar
[26]Urbański, M. and Zdunik, A.. Real analyticity of Hausdorff dimension of finer Julia sets of exponential family. Ergod. Th. & Dynam. Sys. 24(1) (2004), 279–315.CrossRefGoogle Scholar
[27]Weiss, H.. The Lyapunov spectrum for conformal expanding maps and axiom-A surface diffeomorphisms. J. Stat. Phys. 95(3–4) (1999), 615–632.CrossRefGoogle Scholar