Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-27T12:38:20.563Z Has data issue: false hasContentIssue false
  • English
  • Français

Thermodynamics of smooth models of pseudo–Anosov homeomorphisms

Part of: Dynamical systems with hyperbolic behavior Low-dimensional dynamical systems Smooth dynamical systems: general theory Ergodic theory

Published online by Cambridge University Press:  18 May 2021

DOMINIC VECONI
DOMINIC VECONI*
Affiliation:
Department of Mathematics, Penn State University, State College, Pennsylvania16802, USA
*
e-mail: [email protected]
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 develop a thermodynamic formalism for a smooth realization of pseudo-Anosov surface homeomorphisms. In this realization, the singularities of the pseudo-Anosov map are assumed to be fixed, and the trajectories are slowed down so the differential is the identity at these points. Using Young towers, we prove existence and uniqueness of equilibrium states for geometric t-potentials. This family of equilibrium states includes a unique SRB measure and a measure of maximal entropy, the latter of which has exponential decay of correlations and the central limit theorem.

Keywords

nonuniform hyperbolicitypseudo-Anosov diffeomorphismsthermodynamic formalismsmooth ergodic theory

MSC classification

Primary: 37C05: Smooth mappings and diffeomorphisms 37C40: Smooth ergodic theory, invariant measures 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) 37D35: Thermodynamic formalism, variational principles, equilibrium states
Secondary: 37A50: Relations with probability theory and stochastic processes 37E30: Homeomorphisms and diffeomorphisms of planes and surfaces
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 3: Anatole Katok Memorial Issue Part 2: Special Issue of Ergodic Theory and Dynamical Systems , March 2022 , pp. 1284 - 1326
Check for updates
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Footnotes

Dedicated to the memory of Anatole Katok

References

Barreira, L., Pesin, Y.. Nonuniform Hyperbolicity: Dynamics of systems with nonzero Lyapunov exponents (Encyclopedia of Mathematics and Its Applications, 115) . Cambridge University Press, 2007.CrossRefGoogle Scholar
Bowen, R.. Equilibrium states and ergodic theory of Anosov diffeomorphisms (Lecture Notes in Mathematics, 470) . Springer, 1975.CrossRefGoogle Scholar
Buzzi, J., Crovisier, S., and Sarig, O.. Measures of maximal entropy for surface diffeomorphisms. Preprint. 2019, arXiv:1811.02240.Google Scholar
Climenhaga, V., Pesin, Y.. Building thermodynamics for non-uniformly hyperbolic maps. Arnold Math. J. 3(1) (2017), 37–82.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
Fathi, A., Laudenbach, F., and Poénaru, V., Travaux de Thurston sur les surfaces, in Séminaire Orsay. Astérisque 66–67 (1979).Google Scholar
Faure, F., Gouëzel, S., Lanneau, E.. Ruelle spectrum of linear pseudo-Anosov maps. J. Éc. Polytech. Math . 6 (2019), 811–877.CrossRefGoogle Scholar
Gerber, M., Katok, A.. Smooth models of pseudo-Anosov maps. Ann. Sci. Éc. Norm. Supér. (4) 15(4) (1982), 173–204.CrossRefGoogle Scholar
Katok, A.. Bernoulli diffeomorphisms on surfaces. Ann. of Math. (2) , 110(3) (1979), 529–547.CrossRefGoogle Scholar
Ledrappier, F. and Young, L. S., The metric entropy of diffeomorphisms, Bull. Amer. Math. Soc. (N.S.) , 11(2) (1984), 343–346.CrossRefGoogle Scholar
Penner, R.. A construction of pseudo-Anosov homeomorphisms. Trans. Amer. Math. Soc. 310(1) (1988), 179–197.CrossRefGoogle Scholar
Pesin, Y., Senti, S., and Zhang, K.. Thermodynamics of towers of hyperbolic type. Trans. Amer. Math. Soc. 368(12) (2016), 8519–8552.CrossRefGoogle Scholar
Pesin, Y., Senti, S., and Zhang, K.. Thermodynamics of the Katok map (revised version). Ergod. Th. & Dynam. Sys. 39(3) (2019), 764–794.CrossRefGoogle Scholar
Pollicot, M., Weiss, H.. Multifractal analysis of the Lyapunov exponent for continued fraction and Manneville-Pomeau transformations and applications to Diophantine approximations. Comm. Math. Phys. 207 (1999), 145–171.CrossRefGoogle Scholar
Pomeau, Y. and Manneville, P.. Intermittent transition to turbulence in dissipative dynamical systems. Comm. Math. Phys. , 74(2) (1980), 189–197.CrossRefGoogle Scholar
Rodriguez-Hertz, F., Rodriguez-Hertz, M. A., Tahzibi, A., and Ures, R.. Uniqueness of SRB measures for transitive diffeomorphisms on surfaces, Comm. Math. Phys. 306(1) (2011), 35–49.CrossRefGoogle Scholar
Shahidi, F., Zelerowicz, A.. Thermodynamics via inducing, J. Stat. Phys , 175(2) (2019), 351–383.CrossRefGoogle Scholar
Thurston, W.. On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Amer. Math. Soc. (N.S.) , 19(2) (1988), 417–431.CrossRefGoogle Scholar
Veconi, D.. Equilibrium states of almost Anosov diffeomorphisms. Discrete Contin. Dyn. Syst. 40(2) (2020), 767–780.CrossRefGoogle Scholar
Venegeroles, R.. Thermodynamic phase transitions for Pomeau-Manneville maps. Phys. Rev. 86(2) (2012), 021114.Google ScholarPubMed
Wang, T.. Unique equilibrium states, large deviations and Lyapunov spectra for the Katok map. Ergod. Th. & Dynam. Sys. doi: 10.1017/etds.2020.26.2020. Published online 20 March 2020.Google Scholar