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Intrinsic ergodicity via obstruction entropies

Published online by Cambridge University Press:  03 April 2013

VAUGHN CLIMENHAGA  and
DANIEL J. THOMPSON
VAUGHN CLIMENHAGA
Affiliation:
Department of Mathematics, University of Houston, Houston, Texas 77204, USA email [email protected]
DANIEL J. THOMPSON
Affiliation:
Department of Mathematics, The Ohio State University, 100 Math Tower, 231 West 18th Avenue, Columbus, Ohio 43210, USA email [email protected]
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Abstract

Bowen showed that a continuous expansive map with specification has a unique measure of maximal entropy. We show that the conclusion remains true under weaker non-uniform versions of these hypotheses. To this end, we introduce the notions of obstructions to expansivity and specification, and show that if the entropy of such obstructions is smaller than the topological entropy of the map, then there is a unique measure of maximal entropy.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 6 , December 2014 , pp. 1816 - 1831
Copyright
Copyright ©2013 Cambridge University Press 

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References

Buzzi, J. and Fisher, T.. Entropic stability beyond partial hyperbolicity. Preprint, 2011, 27 pp., arXiv:1103.2707.Google Scholar
Brin, M. and Katok, A.. On local entropy. Geometric Dynamics (Rio de Janeiro, 1981) (Lecture Notes in Mathematics, 1007). Springer, Berlin, 1983, pp. 30–38.Google Scholar
Bowen, R.. Some systems with unique equilibrium states. Math. Systems Theory 8 (3) (1974/75), 193–202.CrossRefGoogle Scholar
Buzzi, J.. Intrinsic ergodicity of smooth interval maps. Israel J. Math. 100 (1) (1997), 125–161.Google Scholar
Climenhaga, V.. Topological pressure of simultaneous level sets. Nonlinearity 26 (1) (2013), 241–268.Google Scholar
Climenhaga, V. and Thompson, D. J.. Intrinsic ergodicity beyond specification: $\beta $-shifts, $S$-gap shifts, and their factors. Israel J. Math. 192 (2) (2012), 785–817.CrossRefGoogle Scholar
Hofbauer, F.. On intrinsic ergodicity of piecewise monotonic transformations with positive entropy. Israel J. Math. 34 (3) (1979), 213–237.CrossRefGoogle Scholar
Hofbauer, F.. On intrinsic ergodicity of piecewise monotonic transformations with positive entropy II. Israel J. Math. 38 (1–2) (1981), 107–115.CrossRefGoogle Scholar
Lindenstrauss, E. and Schmidt, K.. Invariant sets and measures of nonexpansive group automorphisms. Israel J. Math. 144 (2004), 29–60.Google Scholar
Pesin, Y. B.. Dimension Theory in Dynamical Systems (Chicago Lectures in Mathematics, Contemporary Views and Applications). University of Chicago Press, Chicago, 1997.Google Scholar
Schmidt, K.. Algebraic coding of expansive group automorphisms and two-sided beta-shifts. Monatsh. Math. 129 (1) (2000), 37–61.Google Scholar
Sidorov, N.. Arithmetic dynamics. Topics in Dynamics and Ergodic Theory (London Mathematical Society Lecture Notes Series, 310). Cambridge University Press, Cambridge, 2003, pp. 145–189.Google Scholar
Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79). Springer, New York, 1982.Google Scholar
Weiss, B.. Intrinsically ergodic systems. Bull. Amer. Math. Soc. 76 (6) (1970), 1266–1269.Google Scholar