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Intrinsic ergodicity of fibered products of one-dimensional chaotic maps. (Ergodicité intrinsèque de produits fibrés d’applications chaotiques unidimensionelles.) (French) Zbl 0917.58018
Summary: We consider a small fibered perturbations of direct products of \(C^\infty\) interval maps with positive entropy. These non-expanding nonlinear multi-dimensional systems are shown to have a non-zero and finite number of invariant and ergodic probability measures with maximal entropy. The proof uses a suitable generalization of the Hofbauer diagram which gives isomorphism (in the sense of entropy) with a countable topological Markov chain having a finite number of irreducible sub-chains. The main point of the proof is to estimate topological entropies using approximations by semi-algebraic sets and semi-continuity results.

MSC:
37A99 Ergodic theory
28D20 Entropy and other invariants
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[1] BOWEN (R.) . - Entropy for group endomorphisms and homogeneous spaces , Trans. Amer. Math. Soc., t. 153, 1971 , p. 401-414. MR 43 #469 | Zbl 0212.29201 · Zbl 0212.29201
[2] BUZZI (J.) . - Entropies et représentation markovienne des applications régulières de l’intervalle . - Thèse, Université Paris-Sud, Orsay, 1995 .
[3] BUZZI (J.) . - Intrinsic Ergodicity of Smooth Interval Maps , Israel J. Math., t. 100, 1997 , p. 125-161. MR 99g:58071 | Zbl 0889.28009 · Zbl 0889.28009
[4] BUZZI (J.) . - Intrinsic Ergodicity of Affine Maps on [0, 1]d , Monat. Math., t. 124, 1997 , p. 97-118. MR 98g:58098 | Zbl 0886.58024 · Zbl 0886.58024
[5] BUZZI (J.) . - Entropy, volume growth and Lyapunov exponents , submitted.
[6] BUZZI (J.) . - Markov extensions for multi-dimensional dynamical systems , submitted. · Zbl 0988.37012
[7] COSTE (M.) . - Ensembles semi-algébriques , Lecture Notes in Mathematics, n^\circ 959, Springer-Verlag, Berlin, 1982 , p. 109-138. MR 84e:14023 | Zbl 0498.14012 · Zbl 0498.14012
[8] DENKER (M.) , GRILLENBERG (C.) , SIGMUND (K.) . - Ergodic theory on compact spaces , Lecture Notes in Mathematics n^\circ 527, Springer-Verlag, 1976 . MR 56 #15879 | Zbl 0328.28008 · Zbl 0328.28008
[9] GROMOV (M.) . - Entropy, homology and semi-algebraic geometry , Séminaire Bourbaki, t. 663, 1985 - 1986 . Numdam | Zbl 0611.58041 · Zbl 0611.58041
[10] GUREVIČ (B.M.) . - Topological entropy of enumerable Markov chains , Soviet Math. Dokl., t. 10, 1969 , p. 911-915. Zbl 0194.49602 · Zbl 0194.49602
[11] GUREVIČ (B.M.) . - Shift entropy and Markov measures in the path space of a denumerable graph , Soviet Math. Dokl., t. 11, 1970 , p. 744-747. Zbl 0217.38101 · Zbl 0217.38101
[12] HIRSCH (M.W.) , PUGH (C.C.) , SHUB (M.) . - Invariant Manifolds , Lecture Notes in Mathematics n^\circ 583, Springer-Verlag, Berlin, 1977 . MR 58 #18595 | Zbl 0355.58009 · Zbl 0355.58009
[13] HOFBAUER (F.) . - On intrinsic ergodicity of piecewise monotonic transformations with positive entropy , Israel J. Math., t. I, 34, 1979 , p. 213-237; t. II, 38, 1981 , p. 107-115. Zbl 0422.28015 · Zbl 0422.28015
[14] KATOK (A.) . - Lyapunov exponents, entropy and periodic orbits for diffeomorphisms , Publ. Math. I.H.E.S., t. 51, 1980 , p. 137-173. Numdam | MR 81i:28022 | Zbl 0445.58015 · Zbl 0445.58015
[15] KELLER (G.) . - Markov extensions, zeta functions, and Fredholm theory for piecewise invertible dynamical systems , Trans. Amer. Math. Soc, t. 314, 1989 , p. 433-497. MR 91c:58062 | Zbl 0686.58027 · Zbl 0686.58027
[16] KELLER (G.) . - Lifting measures to Markov extensions , Mh. Math., t. 108, 1989 , p. 183-200. Article | MR 91b:28011 | Zbl 0712.28008 · Zbl 0712.28008
[17] LEDRAPPIER (F.) , WALTERS (P.) . - A relativised variational principle for continuous transformations , J. London Math. Soc., t. 16, 1977 , p. 568-576. MR 57 #16540 | Zbl 0388.28020 · Zbl 0388.28020
[18] MISIUREWICZ (M.) . - Horseshoes for mappings of the interval , Bull. Acad. Polon. Sci., Ser. Sci. Math., t. 27, 1979 , p. 167-169. MR 81b:58033 | Zbl 0459.54031 · Zbl 0459.54031
[19] MISIUREWICZ (M.) , SZLENK (W.) . - Entropy of piecewise monotone mappings , Studia Math., t. 67, p. 45-63. Article | MR 82a:58030 | Zbl 0445.54007 · Zbl 0445.54007
[20] NEWHOUSE (S.E.) . - Continuity properties of the entropy , Ann. Math., t. 129, 1989 , p. 215-237. MR 90f:58108 | Zbl 0676.58039 · Zbl 0676.58039
[21] WEISS (B.) . - Intrinsically ergodic systems , Bull. Amer. Math. Soc., t. 76, 1970 , p. 1266-1269. Article | MR 42 #1978 | Zbl 0218.28011 · Zbl 0218.28011
[22] YOMDIN (Y.) . - Volume growth and entropy , Israel J. Math., t. 57, 1987 , p. 285-300. MR 90g:58008 | Zbl 0641.54036 · Zbl 0641.54036
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