Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-27T04:14:24.288Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected])
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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