Large entropy implies existence of a maximal entropy measure for interval maps

Discrete and Continuous Dynamical Systems

Volumn 14, Issue 4, 2006, Pages 673-688

  Buzzi, Jérôme ^a   Ruette, Sylvie ^b  

^a ECOLE POLYTECHNIQUE   (France)

^b UNIVERSITÉ PARIS SACLAY   (France)

Author keywords

Interval map; Markov shift; Maximal entropy measure

Indexed keywords

EID: 33645777157     PISSN: 10780947     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcds.2006.14.673     Document Type: Review

References (27)

1. A.M. Blokh, "Spectral expansion" for piecewise monotone mappings of an interval (Russian), Uspekhi Mat. Nauk, 37 no. 3(225) (1982), 175-176.
2. A.M. Blokh, The "spectral" decomposition for one-dimensional maps, Dynamics reported, 1-59, Dynam. Report. Expositions Dynam. Systems (N.S.), 4, Springer, Berlin, 1995.
3. R. Bowen, Entropy-expansive maps, Trans. Amer. Math. Soc., 164 (1972), 323-331.
4. R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414.
5. R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975.
6. J. Buzzi, Entropies et représentations markoviennes des applications régulières de l'intervalle, thèse de l'Université de Paris-Sud, Orsay, 1995.
7. J. Buzzi, Intrinsic ergodicity of smooth interval maps, Israel J. Math., 100 (1997), 125-161.
8. J. Buzzi, Markov extensions for multi-dimensional dynamical systems, Israel J. Math., 112 (1999), 357-380.
9. J. Buzzi, On entropy-expanding maps, Preprint CMAT, Ecole polytechnique, 2000.
10. M. Denker, Measures with maximal entropy, Théorie ergodique (actes Journées Ergodiques, Rennes, 1973/1974), Lecture Notes in Mathematics, Vol. 532, pages 70-112, Springer-Verlag, Berlin-New York, 1976.
11. M. Denker, C. Grillenberger, K. Sigmund, Ergodic theory on compact spaces, Lecture Notes in Mathematics, Vol. 527, Springer-Verlag, Berlin-New York, 1976.
12. B.M. Gurevich, Topological entropy of enumerable Markov chains, Soviet. Math. Dokl., 10 no. 4 (1969), 911-915.
13. B.M. Gurevich, Shift entropy and Markov measures in the path space of a denumerable graph, Soviet. Math. Dokl., 11 (1970), 744-747.
14. B.M. Gurevich and S.V. Savchenko, Thermodynamic formalism for countable symbolic Markov chains (Russian), Uspekhi Mat. Nauk, 53 no. 2 (1998), 3-106;
15. English translation: Russian Math. Surveys, 53 no. 2 (1998), 245-344.
16. F. Hofbauer, β-shifts have unique maximal measure, Monatsh. Math., 85 no. 2 (1978), 189-198.
17. F. Hofbauer, On intrinsic ergodicity of piecewise monotonic transformations with positive entropy, Israel J. Math., 34 (1979), 213-237;
18. F. Hofbauer, On intrinsic ergodicity of piecewise monotonic transformations with positive entropy, Israel J. Math., II. 38 (1981), 107-115.
19. A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51, (1980), 137-173.
20. M. Misiurewicz, Diffeomorphism without any measure with maximal entropy, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 21 (1973), 903-910.
21. M. Misiurewicz, Topological conditional entropy, Studia Math., 55 (1976), 175-200.
22. S. Newhouse, Continuity properties of entropy., Ann. of Math. (2), 129 (1989), 215-235; Erratum: 131 (1990), 409-410.
23. D. Ruelle, An inequality for the entropy of differentiable maps., Bol. Soc. Brasil. Mat., 9 (1978), 83-87.
24. r maps of the interval without maximal measure, Israel J. Math., 127, 253-277, 2002.
25. Y. Takahashi, Isomorphisms of β-automorphisms to Markov automorphisms, Osaka J. Math., 10 (1973), 175-184.
26. D. Vere-Jones, Geometric ergodicity in denumerable Markov chains, Quarterly J. Math., 13 (1962), 7-28.
27. P. Walters, An introduction to ergodic theory, Springer-Verlag, New York, 1982.