Hostname: page-component-74656f9884-54qdv Total loading time: 0 Render date: 2026-11-10T08:23:59.658Z Has data issue: false hasContentIssue false
  • English
  • Français

Maximizing measures for partially hyperbolic systems with compact center leaves

Published online by Cambridge University Press:  05 December 2011

F. RODRIGUEZ HERTZ ,
M. A. RODRIGUEZ HERTZ ,
A. TAHZIBI  and
R. URES
F. RODRIGUEZ HERTZ
Affiliation:
IMERL-Facultad de Ingeniería, Universidad de la República, CC 30 Montevideo, Uruguay (email: frhertz@fing.edu.uy, jana@fing.edu.uy, ures@fing.edu.uy)
M. A. RODRIGUEZ HERTZ
Affiliation:
IMERL-Facultad de Ingeniería, Universidad de la República, CC 30 Montevideo, Uruguay (email: frhertz@fing.edu.uy, jana@fing.edu.uy, ures@fing.edu.uy)
A. TAHZIBI
Affiliation:
Departamento de Matemática, ICMC-USP São Carlos, Caixa Postal 668, 13560-970 São, Carlos-SP, Brazil (email: tahzibi@icmc.sc.usp.br)
R. URES
Affiliation:
IMERL-Facultad de Ingeniería, Universidad de la República, CC 30 Montevideo, Uruguay (email: frhertz@fing.edu.uy, jana@fing.edu.uy, ures@fing.edu.uy)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We obtain the following dichotomy for accessible partially hyperbolic diffeomorphisms of three-dimensional manifolds having compact center leaves: either there is a unique entropy-maximizing measure, this measure has the Bernoulli property and its center Lyapunov exponent is 0, or there are a finite number of entropy-maximizing measures, all of them with non-zero center Lyapunov exponents (at least one with a negative exponent and one with a positive exponent), that are finite extensions of a Bernoulli system. In the first case of the dichotomy, we obtain that the system is topologically conjugated to a rotation extension of a hyperbolic system. This implies that the second case of the dichotomy holds for an open and dense set of diffeomorphisms in the hypothesis of our result. As a consequence, we obtain an open set of topologically mixing diffeomorphisms having more than one entropy-maximizing measure.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 2: Daniel J. Rudolph – in Memoriam , April 2012 , pp. 825 - 839
Copyright
Copyright © Cambridge University Press 2011

References

[1]Avila, A. and Viana, M.. Extremal Lyapunov exponents: an invariance principle and applications. Invent. Math. 181 (2010), 115–178.CrossRefGoogle Scholar
[2]Avila, A., Viana, M. and Wilkinson, A.. Absolute continuity, Lyapunov exponents, and rigidity. Preprint.Google Scholar
[3]Bonatti, C. and Díaz, L. J.. Persistent nonhyperbolic transitive diffeomorphisms. Ann. of Math. (2) 143(2) (1996), 357–396.CrossRefGoogle Scholar
[4]Bowen, R.. Markov partitions for Axiom A diffeomorphisms. Amer. J. Math. 92 (1970), 725–747.CrossRefGoogle Scholar
[5]Brin, M., Burago, D. and Ivanov, S.. Dynamical coherence of partially hyperbolic diffeomorphisms of the 3-torus. J. Mod. Dyn. 3 (2009), 1–11.CrossRefGoogle Scholar
[6]Burns, K., Rodriguez Hertz, F., Rodriguez Hertz, M. A., Talitskaya, A. and Ures, R.. Density of accessibility for partially hyperbolic diffeomorphisms with one-dimensional center. Discrete Contin. Dyn. Syst. 22(1–2) (2008), 75–88.Google Scholar
[7]Buzzi, J., Fisher, T., Sambarino, M. and Vasquez, C.. Maximal entropy measures for certain partially hyperbolic, derived from Anosov systems. Ergod. Th. & Dynam. Sys. to appear.Google Scholar
[8]Carrasco, P.. Compact dynamical foliations. PhD Thesis, University of Toronto, 2010.Google Scholar
[9]Cowieson, W. and Young, L.-S.. SRB measures as zero-noise limits. Ergod. Th. & Dynam. Sys. 25(4) (2005), 1115–1138.CrossRefGoogle Scholar
[10]Díaz, L. J. and Fisher, T.. Symbolic extensions for partially hyperbolic diffeomorphisms. Discrete Contin. Dyn. Syst. 29 (2011), 1419–1441.CrossRefGoogle Scholar
[11]Didier, P.. Stability of accessibility. Ergod. Th. & Dynam. Sys. 23(6) (2003), 1717–1731.CrossRefGoogle Scholar
[12]Dolgopyat, D.. On mixing properties of compact group extensions of hyperbolic systems. Israel J. Math. 130 (2002), 157–205.CrossRefGoogle Scholar
[13]Epstein, D. B. A.. Periodic flow on three-manifolds. Ann. of Math. (2) 95(1) (1972), 66–82.CrossRefGoogle Scholar
[14]Franks, J.. Anosov diffeomorphisms. Global Analysis (Proceedings of Symposia in Pure Mathematics, XIV, Berkeley, CA, 1968). American Mathematical Society, Providence, RI, 1970, pp. 61–93.CrossRefGoogle Scholar
[15]Hatcher, A.. Notes on basic 3-manifold topology. Hatcher’s webpage at Cornell Math. Dep. http://www.math.cornell.edu/∼hatcher/3M/3M.pdf.Google Scholar
[16]Hammerlindl, A.. Leaf conjugacies of the torus. Ergod. Th. & Dynam. Sys. to appear and PhD Thesis, University of Toronto, 2009.Google Scholar
[17]Hua, Y., Saghin, R. and Xia, Z.. Topological entropy and partially hyperbolic diffeomorphisms. Ergod. Th. & Dynam. Sys. 28(3) (2008), 843–862.CrossRefGoogle Scholar
[18]Ledrappier, F. and Walters, P.. A relativised variational principle for continuous transformations. J. Lond. Math. Soc. (2) 16(3) (1977), 568–576.CrossRefGoogle Scholar
[19]Mañé, R.. Contributions to the stability conjecture. Topology 17 (1978), 383–396.CrossRefGoogle Scholar
[20]Margulis, G.. On Some Aspects of the Theory of Anosov Systems (Springer Monographs in Mathematics, VII). Springer, New York, 2004, with a survey by Richard Sharp: periodic orbits of hyperbolic flows.CrossRefGoogle Scholar
[21]Misiurewicz, M.. Diffeomorphism without any measure with maximal entropy. Bull. Acad. Polon. Sci., Sér. Sci. Math., Astr. et Phys. 21 (1973), 903–910.Google Scholar
[22]Nassiri, M. and Pujals, E.. Robust transitivity in Hamiltonian dynamics. Ann. Sci. ENS to appear.Google Scholar
[23]Newhouse, S.. Continuity properties of entropy. Ann. of Math. (2) 129 (1989), 215–235.CrossRefGoogle Scholar
[24]Newhouse, S. and Young, L. S.. Dynamics of certain skew products. Geometric Dynamics (Rio de Janeiro, 1981) (Lecture Notes in Mathematics, 1007). Springer, Berlin, 1983, pp. 611–629.CrossRefGoogle Scholar
[25]Niţică, V. and Török, A.. An open dense set of stably ergodic diffeomorphisms in a neighborhood of a non-ergodic one. Topology 40 (2001), 259–278.CrossRefGoogle Scholar
[26]Parwani, K.. On 3-manifolds that support partially hyperbolic diffeomorphisms. Nonlinearity 23(3) (2010), 589–606.CrossRefGoogle Scholar
[27]Rodriguez Hertz, F., Rodriguez Hertz, M. and Ures, R.. Partial hyperbolicity and ergodicity in dimension three. J. Mod. Dyn. 2 (2008), 187–208.CrossRefGoogle Scholar
[28]Rodriguez Hertz, F., Rodriguez Hertz, M. A. and Ures, R.. A non-dynamical coherent example in 𝕋3. in preparation.Google Scholar
[29]Rudolph, D. J.. Classifying the isometric extensions of a Bernoulli shift. J. Anal. Math. 34 (1978), 36–60.CrossRefGoogle Scholar
[30]Ruelle, D. and Wilkinson, A.. Absolutely singular dynamical foliations. Comm. Math. Phys. 219 (2001), 481–487.CrossRefGoogle Scholar
[31]Shub, M.. Topologically transitive diffeomorphisms on 𝕋4. Lecture Notes on Math. 206 (1971), 39.CrossRefGoogle Scholar
[32]Steenrod, N.. The Topology of Fibre Bundles (Princeton Mathematical Series, 14). Princeton University Press, Princeton, NJ, 1951.CrossRefGoogle Scholar
[33]Nassiri, M. and Tahzibi, A.. Ergodic properties of partially hyperbolic sets. in preparation.Google Scholar
[34]Ures, R.. Intrinsic ergodicity of partially hyperbolic diffeomorphisms with hyperbolic linear part. Proc. Amer. Math. Soc. to appear.Google Scholar
[35]Weiss, B.. Intrinsically ergodic systems. Bull. Amer. Math. Soc. 76 (1970), 1266–1269.CrossRefGoogle Scholar