Hostname: page-component-6587cd75c8-vfwnz Total loading time: 0 Render date: 2025-04-24T04:33:21.678Z Has data issue: false hasContentIssue false
  • English
  • Français

Thermodynamical u-formalism I: measures of maximal u-entropy for maps that factor over Anosov

Part of: Ergodic theory Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  27 February 2023

RAUL URES Open the ORCID record for RAUL URES [Opens in a new window] ,
MARCELO VIANA Open the ORCID record for MARCELO VIANA [Opens in a new window] ,
FAN YANG Open the ORCID record for FAN YANG [Opens in a new window]  and
JIAGANG YANG Open the ORCID record for JIAGANG YANG [Opens in a new window]
RAUL URES
Affiliation:
Department of Mathematics, Southern University of Science and Technology, Shenzhen, Guangdong, China SUSTech International Center for Mathematics, Shenzhen, Guangdong, China (e-mail: [email protected])
MARCELO VIANA
Affiliation:
IMPA, Est. D. Castorina 110, 22460-320 Rio de Janeiro, Brazil (e-mail: [email protected])
FAN YANG
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, MI, USA (e-mail: [email protected])
JIAGANG YANG*
Affiliation:
Departamento de Geometria, Instituto de Matemática e Estatística, Universidade Federal Fluminense, Niterói, Brazil
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We construct measures of maximal u-entropy for any partially hyperbolic diffeomorphism that factors over an Anosov torus automorphism and has mostly contracting center direction. The space of such measures has finite dimension, and its extreme points are ergodic measures with pairwise disjoint supports.

Keywords

partially hyperbolic diffeomorphismu-entropymostly contracting center

MSC classification

Primary: 37A35: Entropy and other invariants, isomorphism, classification 37D30: Partially hyperbolic systems and dominated splittings 37D35: Thermodynamic formalism, variational principles, equilibrium states
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 44 , Issue 1 , January 2024 , pp. 290 - 333
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Arnaud, M.-C., Bonatti, C. and Crovisier, S.. Dynamiques symplectiques génériques. Ergod. Th. & Dynam. Sys. 25 (2005), 1401–1436.CrossRefGoogle Scholar
Alves, J. F., Bonatti, C. and Viana, M.. SRB measures for partially hyperbolic systems whose central direction is mostly expanding. Invent. Math. 140 (2000), 351–398.CrossRefGoogle Scholar
Andersson, M.. Robust ergodic properties in partially hyperbolic dynamics. Trans. Amer. Math. Soc. 362 (2010), 1831–1867.CrossRefGoogle Scholar
Avila, A. and Viana, M.. Extremal Lyapunov exponents: an invariance principle and applications. Invent. Math. 181 (2010), 115–189.CrossRefGoogle Scholar
Avila, A., Viana, M. and Wilkinson, A.. Absolute continuity, Lyapunov exponents, and rigidity II: systems with compact center leaves. Ergod. Th. & Dynam. Sys. 42 (2022), 437–490.CrossRefGoogle Scholar
Bochi, J. and Bonatti, C.. Perturbation of the Lyapunov spectra of periodic orbits. Proc. Lond. Math. Soc. (3) 105 (2012), 1–48.CrossRefGoogle Scholar
Bonatti, C. and Crovisier, S.. Récurrence et généricité. Invent. Math. 158 (2004), 33–104.CrossRefGoogle Scholar
Buzzi, J., Crovisier, S. and Fisher, T.. The entropy of ${c}^1$ -diffeomorphisms without a dominated splitting. Trans. Amer. Math. Soc. 370 (2018), 6685–6734.CrossRefGoogle Scholar
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
Bonatti, C., Díaz, L. J. and Viana, M.. Dynamics Beyond Uniform Hyperbolicity (Encyclopaedia of Mathematical Sciences, 102). Springer-Verlag, Berlin, 2005.Google Scholar
Bonatti, C., Li, M. and Yang, D.. On the existence of attractors. Trans. Amer. Math. Soc. 365 (2013), 1369–1391.CrossRefGoogle Scholar
Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in Mathematics, 470). Springer-Verlag, Berlin, 1975.CrossRefGoogle Scholar
Bowen, R. and Ruelle, D.. The ergodic theory of Axiom A flows. Invent. Math. 29 (1975), 181–202.CrossRefGoogle Scholar
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 (2008), 75–88.Google Scholar
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
Burns, K. and Wilkinson, A.. On the ergodicity of partially hyperbolic systems. Ann. of Math. (2) 171 (2010), 451–489.CrossRefGoogle Scholar
Climenhaga, V., Pesin, Y. and Zelerowicz, A.. Equilibrium measures for some partially hyperbolic systems. J. Mod. Dyn. 16 (2020), 155–205.CrossRefGoogle Scholar
Didier, P.. Stability of accessibility. Ergod. Th. & Dynam. Sys. 23(6) (2003), 1717–1731.CrossRefGoogle Scholar
Dinaburg, E.. A correlation between topological entropy and metric entropy. Dokl. Akad. Nauk SSSR 190 (1970), 19–22.Google Scholar
Dinaburg, E.. A connection between various entropy characterizations of dynamical systems. Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 324–366.Google Scholar
Dolgopyat, D.. On dynamics of mostly contracting diffeomorphisms. Comm. Math. Phys. 213 (2000), 181–201.CrossRefGoogle Scholar
Dolgopyat, D., Viana, M. and Yang, J.. Geometric and measure-theoretical structures of maps with mostly contracting center. Comm. Math. Phys. 341 (2016), 991–1014.CrossRefGoogle Scholar
Franks, J.. Anosov diffeomorphisms. Global Analysis (Proceedings of Symposia in Pure Mathematics, Vol. XIV, Berkeley, CA, 1968). Eds. S.-S. Chern and S. Smale. American Mathematical Society, Providence, RI, 1970, pp. 61–93.Google Scholar
Gan, S.. A generalized shadowing lemma. Discrete Contin. Dyn. Syst. 8 (2002), 627–632.CrossRefGoogle Scholar
Gan, S., Li, M., Viana, M. and Yang, J.. Partially volume expanding diffeomorphisms. Ann. Henri Poincaré 22 (2021), 331–346.CrossRefGoogle Scholar
Goodman, T.. Relating topological entropy and measure entropy. Bull. Lond. Math. Soc. 3 (1971), 176–180.CrossRefGoogle Scholar
Goodwin, G.. Optimal input signals for nonlinear-system identification. Proc. Inst. Electr. Engrs. 118 (1971), 922–926.CrossRefGoogle Scholar
Gourmelon, N.. Generation of homoclinic tangencies by ${C}^1$ -perturbations. Discrete Contin. Dyn. Syst. 26 (2010), 1–42.CrossRefGoogle Scholar
Gourmelon, N.. A Franks lemma that preserves invariant manifolds. Ergod. Th. & Dynam. Sys. 36 (2016), 1167–1203.CrossRefGoogle Scholar
Hu, H., Hua, Y. and Wu, W.. Unstable entropies and variational principle for partially hyperbolic diffeomorphisms. Adv. Math. 321 (2017), 31–68.CrossRefGoogle Scholar
Hiraide, K.. Expansive homeomorphisms with the pseudo-orbit tracing property of $n$ -tori. J. Math. Soc. Japan 41 (1989), 357–389.CrossRefGoogle Scholar
Hammerlindl, A. and Potrie, R.. Pointwise partial hyperbolicity in three-dimensional nilmanifolds. J. Lond. Math. Soc. (2) 89 (2014), 853–875.CrossRefGoogle Scholar
Hirsch, M., Pugh, C. and Shub, M.. Invariant Manifolds (Lecture Notes in Mathematics, 583). Springer-Verlag, Berlin, 1977.CrossRefGoogle Scholar
Hu, H., Wu, W. and Zhu, Y.. Unstable pressure and u-equilibrium states for partially hyperbolic diffeomorphisms. Ergod. Th. & Dynam. Sys. 41 (2021), 3336–3362.CrossRefGoogle Scholar
Katok, A.. Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Publ. Math. Inst. Hautes Études Sci. 51 (1980), 137–173.CrossRefGoogle Scholar
Ledrappier, F.. Propriétés ergodiques des mesures de Sinaï. Publ. Math. Inst. Hautes Études Sci. 59 (1984), 163–188.CrossRefGoogle Scholar
Liao, S.-T.. On $\left(\eta, d\right)$ -contractible orbits of vector fields. J. Systems Sci. Math. Sci. 2 (1989), 193–227.Google Scholar
Ledrappier, F. and Strelcyn, J.-M.. A proof of the estimation from below in Pesin’s entropy formula. Ergod. Th. & Dynam. Sys. 2 (1983), 203–219, 1982.CrossRefGoogle Scholar
Liao, G., Viana, M. and Yang, J.. The entropy conjecture for diffeomorphisms away from tangencies. J. Eur. Math. Soc. (JEMS) 15(6) (2013), 2043–2060.Google Scholar
Ledrappier, F. and Walters, P.. A relativised variational principle for continuous transformations. J. Lond. Math. Soc. (2) 16 (1977), 568–576.CrossRefGoogle Scholar
Ledrappier, F. and Young, L.-S.. The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin’s entropy formula. Ann. of Math. (2) 122 (1985), 509–539.CrossRefGoogle Scholar
Ledrappier, F. and Young, L.-S.. The metric entropy of diffeomorphisms. II. Relations between entropy, exponents and dimension. Ann. of Math. (2) 122 (1985), 540–574.CrossRefGoogle Scholar
Mañé, R.. Contributions to the stability conjecture. Topology 17 (1978), 383–396.CrossRefGoogle Scholar
Margulis, G. A.. Certain measures that are connected with Anosov flows on compact manifolds. Funktsional. Anal. i Prilozhen. 4 (1970), 62–76 (in Russian).CrossRefGoogle Scholar
Newhouse, S.. Topological entropy and Hausdorff dimension for area preserving diffeomorphisms of surfaces. Dynamical Systems, Vol. III—Warsaw (Astérisque, 51). Société mathématique de France, Paris, 1978, pp. 323–334.Google Scholar
Palis, J. and de Melo, W.. Geometric Theory of Dynamical Systems: An Introduction. Springer-Verlag, New York, 1982, translated from the Portuguese by A. K. Manning.CrossRefGoogle Scholar
Pesin, Y. B.. Characteristic Lyapunov exponents and smooth ergodic theory. Russian Math. Surveys 324 (1977), 55–114.CrossRefGoogle Scholar
Plykin, R. V.. Hyperbolic attractors of diffeomorphisms. Uspekhi Mat. Nauk 35 (1980), 94–104. in English translation: Russ. Math. Survey 35(3) (1980), 109–121.Google Scholar
Potrie, R.. Partial hyperbolicity and foliations in ${T}^3$ . J. Mod. Dyn. 9 (2015), 81–121.CrossRefGoogle Scholar
Pesin, Y. and Sinai, Y.. Gibbs measures for partially hyperbolic attractors. Ergod. Th. & Dynam. Sys. 2 (1982), 417–438.CrossRefGoogle Scholar
Pujals, E. and Sambarino, M.. Homoclinic tangencies and hyperbolicity for surface diffeomorphisms. Ann. of Math. (2) 151 (2000), 961–1023.CrossRefGoogle Scholar
Pugh, C.. The ${C}^{1+\alpha }$ hypothesis in Pesin theory. Publ. Math. Inst. Hautes Études Sci. 59 (1984), 143–161.CrossRefGoogle Scholar
Rodriguez-Hertz, F., Rodriguez-Hertz, M. A., Tahzibi, A. and Ures, R.. Maximizing measures for partially hyperbolic systems with compact center leaves. Ergod. Th. & Dynam. Sys. 32 (2012), 825–839.CrossRefGoogle Scholar
Rokhlin, V. A.. Lectures on the entropy theory of measure-preserving transformations. Russian Math. Surveys 22(5) (1967), 1–52, Transl. from Uspekhi Mat. Nauk 22(5) (1967), 3–56.CrossRefGoogle Scholar
Ruelle, D.. A measure associated with Axiom A attractors. Amer. J. Math. 98 (1976), 619–654.CrossRefGoogle Scholar
Ruelle, D.. An inequality for the entropy of differentiable maps. Bull. Braz. Math. Soc. (N.S.) 9 (1978), 83–87.CrossRefGoogle Scholar
Sinai, Y.. Gibbs measures in ergodic theory. Russian Math. Surveys 27 (1972), 21–69.CrossRefGoogle Scholar
Smale, S.. Differentiable dynamical systems. Bull. Amer. Math. Soc. (N.S.) 73 (1967), 747–817.CrossRefGoogle Scholar
Saghin, R. and Xia, Z.. Geometric expansion, Lyapunov exponents and foliations. Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), 689–704.CrossRefGoogle Scholar
Tahzibi, A. and Yang, J.. Invariance principle and rigidity of high entropy measures. Trans. Amer. Math. Soc. 371 (2019), 1231–1251.CrossRefGoogle Scholar
Ures, R.. Intrinsic ergodicity of partially hyperbolic diffeomorphisms with a hyperbolic linear part. Proc. Amer. Math. Soc. 140 (2012), 1973–1985.CrossRefGoogle Scholar
Ures, R., Viana, M. and Yang, J.. Maximal entropy measures of diffeomorphisms of circle fiber bundles. J. Lond. Math. Soc. (2) 103 (2021), 1016–1034.CrossRefGoogle Scholar
Ures, R., Viana, M., Yan, F. and Yang, J.. Fast loss of memory for measures of maximal $u$ -entropy of maps with $c$ -mostly contracting center, to appear.Google Scholar
Viana, M.. Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents. Ann. of Math. (2) 167 (2008), 643–680.CrossRefGoogle Scholar
Viana, M. and Oliveira, K.. Foundations of Ergodic Theory (Cambridge Studies in Advanced Mathematics, 151). Cambridge University Press, Cambridge, 2016.Google Scholar
Viana, M. and Yang, J.. Physical measures and absolute continuity for one-dimensional center direction. Ann. Inst. H. Poincaré Anal. Non Linéaire 30 (2013), 845–877.CrossRefGoogle Scholar
Viana, M. and Yang, J.. Measure-theoretical properties of center foliations. Modern Theory of Dynamical Systems (Contemporary Mathematics, 692). Eds. A. Katok, Y. Pesin, and F. Rodriguez Hertz. American Mathematical Society, Providence, RI, 2017, pp. 291–320.CrossRefGoogle Scholar
Viana, M. and Yang, J.. Continuity of Lyapunov exponents in the ${C}^0$ topology. Israel J. Math. 229 (2019), 461–485.CrossRefGoogle Scholar
Walters, P.. A variational principle for the pressure of continuous transformations. Amer. J. Math. 97 (1975), 937–971.CrossRefGoogle Scholar
Wen, L.. Homoclinic tangencies and dominated splittings. Nonlinearity 15 (2002), 1445–1469.CrossRefGoogle Scholar
Yang, J.. ${\mathrm{C}}^1$ dynamics far from tangencies. PhD Thesis, IMPA, 2008. www.preprint.impa.br.Google Scholar
Yang, J.. Cherry flow: physical measures and perturbation theory. Ergod. Th. & Dynam. Sys. 37 (2017), 2671–2688.CrossRefGoogle Scholar