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Maximal entropy measures for certain partially hyperbolic, derived from Anosov systems

Published online by Cambridge University Press:  10 June 2011

J. BUZZI ,
T. FISHER ,
M. SAMBARINO  and
C. VÁSQUEZ
J. BUZZI
Affiliation:
C.N.R.S. and Département de Mathématiques, Université Paris-Sud, 91405 Orsay, France (email: [email protected])
T. FISHER
Affiliation:
Department of Mathematics, Brigham Young University, Provo, UT 84602, USA (email: [email protected])
M. SAMBARINO
Affiliation:
CMAT-Facultad de Ciencias, U. de la Republica, Montevideo, Uruguay (email: [email protected])
C. VÁSQUEZ
Affiliation:
Instituto de Matemática, Pontificia Universidad Católica de Valparaíso, Valparaíso, Chile (email: [email protected])
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Abstract

We show that a class of robustly transitive diffeomorphisms originally described by Mañé are intrinsically ergodic. More precisely, we obtain an open set of diffeomorphisms which fail to be uniformly hyperbolic and structurally stable, but nevertheless have the following stability with respect to their entropy. Their topological entropy is constant and they each have a unique measure of maximal entropy with respect to which periodic orbits are equidistributed. Moreover, equipped with their respective measure of maximal entropy, these diffeomorphisms are pairwise isomorphic. We show that the method applies to several classes of systems which are similarly derived from Anosov, i.e. produced by an isotopy from an Anosov system, namely, a mixed Mañé example and one obtained through a Hopf bifurcation.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 1 , February 2012 , pp. 63 - 79
Copyright
Copyright © Cambridge University Press 2011

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References

[1]Abraham, R. and Smale, S.. Non-genericity of Ω-stability. Proc. Sympos. Pure Math. 14 (1970), 5–8.CrossRefGoogle Scholar
[2]Bertin, M. J., Decomps-Guilloux, A., Grandet-Hugot, M., Pathiaux-Delefosse, M. and Schreiber, J. P.. Pisot and Salem Numbers. Birkhäuser, Basel, 1992.CrossRefGoogle Scholar
[3]Bonatti, C., Díaz, L. J. and Pujals, E.. A C 1 generic dichotomy for diffeomorphisms; weak forms of hyperbolicity or infinitely many sinks or sources. Ann. of Math. (2) 158 (2003), 355–418.CrossRefGoogle Scholar
[4]Bonatti, C., Díaz, L. J. and Viana, M.. Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective (Encyclopaedia of Mathematical Sciences, 102). Springer, Berlin, 2005.Google Scholar
[5]Bonatti, C. and Viana, M.. SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Israel J. Math. 115 (2000), 157–193.CrossRefGoogle Scholar
[6]Bowen, R.. Entropy for group endomorphisms and homogeneous spaces. Trans. Amer. Math. Soc. 153 (1971), 401–414.CrossRefGoogle Scholar
[7]Bowen, R.. Periodic points and measures for Axiom A diffeomorphisms. Trans. Amer. Math. Soc. 154 (1971), 377–397.Google Scholar
[8]Bowen, R.. Some systems with unique equilibrium states. Math. Syst. Theory 8(3) (1974/75), 193–202.CrossRefGoogle Scholar
[9]Bowen, R. and Ruelle, D.. The ergodic theory of Axiom A flows. Invent. Math. 29(3) (1975), 181–202.CrossRefGoogle Scholar
[10]Buzzi, J. and Fisher, T.. Entropic stability beyond partial hyperbolicity, available athttp://arxiv.org/abs/1103.2707.Google Scholar
[11]Carvalho, M.. Sinai–Ruelle–Bowen measures for N-dimensional [N dimensions] derived from Anosov diffeomorphisms. Ergod. Th. & Dynam. Syst. 13(1) (1993), 21–44.CrossRefGoogle Scholar
[12]Díaz, L. J., Pujals, E. and Ures, R.. Partial hyperbolicity and robust transitivity. Acta Math. 183 (1999), 1–43.CrossRefGoogle Scholar
[13]Hirsch, M. W., Pugh, C. and Shub, M.. Invariant Manifolds (Lecture Notes in Mathematics, 583). Springer, Berlin–New York, 1977.CrossRefGoogle Scholar
[14]Hua, Y., Saghin, R. and Xia, Z.. Topological entropy and partially hyperbolic diffeomorphisms. Ergod. Th. & Dynam. Sys. 28 (2008), 843–862.CrossRefGoogle Scholar
[15]Kan, I.. Open sets of diffeomorphisms having two attractors each with an everywhere dense basin. Bull. Amer. Math. Soc. 31 (1994), 68–74.CrossRefGoogle Scholar
[16]Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
[17]Ledrappier, F. and Walters, P.. A relativized variational principle for continuous transformaitons. J. Lond. Math. Soc. 16 (1977), 568–576.CrossRefGoogle Scholar
[18]Mañé, R.. Contributions to the stability conjecture. Topology 17 (1978), 383–396.CrossRefGoogle Scholar
[19]McSwiggen, P.. Diffeomorphisms of the torus with wandering domains. Proc. Amer. Math. Soc. 117(4) (1993), 1175–1186.CrossRefGoogle Scholar
[20]Misiurewicz, M.. Diffeomorphism without any measures with maximal entropy. Bull. Acad. Polon. Sci. Sèr. Sci. Math. Astronom. Phys. 21 (1973), 903–910.Google Scholar
[21]Newhouse, S.. Continuity properties of entropy. Ann. of Math. (2) 129 (1989), 215–235; Corrections: Ann. of Math. (2) 131 (1990), 409–410.CrossRefGoogle Scholar
[22]Newhouse, S. and Young, L.-S.. Dynamics of Certain Skew Products (Lecture Notes in Mathematics, 1007). Springer, Berlin, 1983, pp. 611–629.Google Scholar
[23]Palis, J.. A global perspective for non-conservative dynamics. Ann. Inst. H. Poincaré Anal. Non Linéaire 22(4) (2005), 485–507.CrossRefGoogle Scholar
[24]Robinson, C.. Dynamical Systems Stability, Symbolic Dynamics, and Chaos. CRC Press, Boca Raton, FL, 1999.Google Scholar
[25]Rodriguez-Hertz, F., Rodriguez-Hertz, M.-A., Tahzibi, A. and Ures, R.. Maximizing measures for partially hyperbolic systems with compact center leaves, available at http://arxiv.org/abs/1010.3372.Google Scholar
[26]Shub, M.. Global Stability of Dynamical Systems. Springer, New York, 1987.CrossRefGoogle Scholar
[27]Smyth, C.. The conjugates of algebraic integers. Advanced Problem, Vol. 5931, Amer. Math. Monthly 82 (1975), 86.CrossRefGoogle Scholar
[28]Wilkinson, A.. Stable ergodicity of the time-one map of a geodesic flow. Ergod. Th. & Dynam. Syst. 18(6) (1998), 1545–1587.CrossRefGoogle Scholar