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Existence of a measurable saturated compensation function between subshifts and its applications

Published online by Cambridge University Press:  13 September 2010

YUKI YAYAMA
YUKI YAYAMA*
Affiliation:
Centro de Modelamiento Matemático, Universidad de Chile, Av. Blanco Encalada 2120, Piso 7, Santiago, Chile (email: yyayama@dim.uchile.cl)
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Abstract

We show the existence of a bounded Borel measurable saturated compensation function for any factor map between subshifts. As an application, we find the Hausdorff dimension and measures of full Hausdorff dimension for a compact invariant set of an expanding non-conformal map on the torus given by an integer-valued diagonal matrix. These problems were studied in [23] for a compact invariant set whose symbolic representation is a shift of finite type under the condition of the existence of a saturated compensation function. By using the ergodic equilibrium states of a constant multiple of a Borel measurable compensation function, we extend the results to the general case where this condition might not hold, presenting a formula for the Hausdorff dimension for a compact invariant set whose symbolic representation is a subshift and studying invariant ergodic measures of full dimension. We study uniqueness and properties of such measures for a compact invariant set whose symbolic representation is a topologically mixing shift of finite type.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 5 , October 2011 , pp. 1563 - 1589
Copyright
Copyright © Cambridge University Press 2010

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References

[1]Barral, J. and Feng, D.-J.. Weighted thermodynamic formalism and applications, arXiv:0909.4247v1.Google Scholar
[2]Barreira, L.. Nonadditive thermodynamic formalism: equilibrium and Gibbs measures. Discrete Contin. Dyn. Syst. 16 (2006), 279–305.CrossRefGoogle Scholar
[3]Bedford, T.. Crinkly curves, Markov partitions and box dimension in self-similar sets. PhD Thesis, University of Warwick, 1984.Google Scholar
[4]Boyle, M. and Tuncel, S.. Infinite-to-one codes and Markov measures. Trans. Amer. Math. Soc. 285 (1984), 657–684.CrossRefGoogle Scholar
[5]Cao, Y.-L., Feng, D.-J. and Huang, W.. The thermodynamic formalism for sub-additive potentials. Discrete Contin. Dyn. Syst. 20 (2008), 639–657.CrossRefGoogle Scholar
[6]Feng, D.-J. and Huang, W.. Lyapunov spectrum of asymptotically sub-additive potentials. Comm. Math. Phys. 297 (2010), 1–43.CrossRefGoogle Scholar
[7]Feng, D.-J.. Equilibrium states for factor maps between subshifts, arXiv:0909.4250v1. Adv. Math., to appear.Google Scholar
[8]Gatzouras, D. and Peres, Y.. The variational principle for Hausdorff dimension: a survey. Ergodic Theory of Zd Actions (London Mathematical Society Lecture Notes, 228). Cambridge University Press, Cambridge, 1996, pp. 113–126.Google Scholar
[9]Kenyon, R. and Peres, Y.. Hausdorff dimensions of sofic affine-invariant sets. Israel J. Math. 94 (1996), 157–178.CrossRefGoogle Scholar
[10]Kenyon, R. and Peres, Y.. Measures of full dimension on affine-invariant sets. Ergod. Th. & Dynam. Sys. 16 (1996), 307–323.CrossRefGoogle Scholar
[11]Ledrappier, F. and Walters, P.. A relativised variational principle for continuous transformations. J. Lond. Math. Soc. (2) 16 (1977), 568–576.CrossRefGoogle Scholar
[12]Lind, D. and Marcus, B.. An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
[13]McMullen, C.. The Hausdorff dimension of general Sierpinski carpets. Nagoya Math. J. 96 (1984), 1–9.CrossRefGoogle Scholar
[14]Mummert, A.. The thermodynamic formalism for almost-additive sequences. Discrete Contin. Dyn. Syst. 16 (2006), 435–454.CrossRefGoogle Scholar
[15]Olivier, E.. Uniqueness of the measure with full dimension on sofic affine-invariant subsets of the 2-torus. Ergod. Th. & Dynam. Sys., available online doi:10.1017/SO143385709000546.CrossRefGoogle Scholar
[16]Petersen, K., Quas, A. and Shin, S.. Measures of maximal relative entropy. Ergod. Th. & Dynam. Sys. 23 (2003), 207–223.CrossRefGoogle Scholar
[17]Petersen, K. and Shin, S.. On the definition of relative pressure for factor maps on shifts of finite type. Bull. Lond. Math. Soc. 37 (2005), 601–612.CrossRefGoogle Scholar
[18]Shin, S.. Relative entropy functions for factor maps between subshifts. Trans. Amer. Math. Soc. 358 (2005), 2205–2216.CrossRefGoogle Scholar
[19]Shin, S.. An example of a factor map without a saturated compensation function. Ergod. Th. & Dynam. Sys. 21 (2001), 1855–1866.CrossRefGoogle Scholar
[20]Shin, S.. Measures that maximize weighted entropy for factor maps between subshifts of finite type. Ergod. Th. & Dynam. Sys. 21 (2001), 1249–1272.CrossRefGoogle Scholar
[21]Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79). Springer, New York, 1982.CrossRefGoogle Scholar
[22]Walters, P.. Relative pressure, relative equilibrium states, compensation functions and many-to-one codes between subshifts. Trans. Amer. Math. Soc. 296 (1986), 1–31.CrossRefGoogle Scholar
[23]Yayama, Y.. Dimensions of compact invariant sets of some expanding maps. Ergod. Th. & Dynam. Sys. 29 (2009), 281–315.CrossRefGoogle Scholar