Advertisement

Journal of Statistical Physics

, Volume 175, Issue 2, pp 351–383 | Cite as

Thermodynamics via Inducing

  • Farruh Shahidi
  • Agnieszka ZelerowiczEmail author
Article
  • 55 Downloads

Abstract

We consider continuous maps \(f:X\rightarrow X\) on compact metric spaces admitting inducing schemes of hyperbolic type introduced in Pesin et al. (Trans Amer Math Soc 368(12):8519–8552, 2016) as well as the induced maps \(\tilde{f}:\tilde{X}\rightarrow \tilde{X}\) and the associated tower maps \(\hat{f}:\hat{X} \rightarrow \hat{X}\). For a certain class of potential functions \(\varphi \) on X, which includes all Hölder continuous functions, we establish thermodynamic formalism for each of the above three systems and we describe some relations between the corresponding equilibrium measures. Furthermore we study ergodic properties of these equilibrium measures including the Bernoulli property, decay of correlations, and the Central Limit Theorem. Finally, we prove analyticity of the pressure function for the three systems.

Keywords

Thermodynamic formalism Inducing schemes 

Notes

Acknowledgements

We would like to thank our advisor, Yakov Pesin, for posing the problem and for valuable suggestions.

References

  1. 1.
    Aaronson, J., Denker, M.: Local limit theorems for partial sums of stationary sequences generated by Gibbs–Markov maps. Stoch. Dyn. 1(2), 193–237 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bareira, L., Pesin, Y.: Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents. Encyclopedia of Mathematics and Its Applications. Cambridge University Press, New York (2007)CrossRefzbMATHGoogle Scholar
  3. 3.
    Bochner, S., Martin, W.T.: Several Complex Variables. Princeton University Press, Princeton (1948)zbMATHGoogle Scholar
  4. 4.
    Climenhaga, V., Luzzatto, S., Pesin, Y.: SRB measures and Young towers for surface diffeomorphisms, PSU preprintGoogle Scholar
  5. 5.
    Daon, Y.: Bernoullicity of equilibrium measures on countable Markov shifts. Dis. Cont. Dyn. Sys. 33(9), 4003–4015 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gouëzel, S.: Sharp polynomial estimates for the decay of correlations. Israel J. Math. 139, 29–65 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kato, T.: Perturbation Theory of Linear Operators. Springer, Berlin (1966)CrossRefzbMATHGoogle Scholar
  8. 8.
    Katok, A.: Bernoulli diffeomorphisms on surfaces. Ann. Math. 110, 529–547 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems. Encyclopedia of Mathematics and its Applications, vol. 54. Cambridge University Press, Cambridge (1995)CrossRefzbMATHGoogle Scholar
  10. 10.
    Mauldin, R.D., Urbanski, M.: Gibbs states on the symbolic space over an infinite alphabet. Israel J. Math. 125, 93–130 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Melbourne, I., Terhesiu, D.: Decay of correlations for non-uniformly expanding systems with general return times. Ergod. Theory Dyn. Syst. 34, 893–918 (2014).  https://doi.org/10.1017/etds.2012.158 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ornstein, D.: Factors of Bernoulli shifts are Bernoulli shifts. Adv. Math. 5, 349–364 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Pesin, Y., Senti, S.: Equilibrium measures for maps with inducing schemes. J. Mod. Dyn. 2(3), 397–430 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Pesin, Y., Senti, S.: Thermodynamics of the Katok map. Ergod. Theory Dyn. Syst. 39, 764–794 (2017).  https://doi.org/10.1017/etds.2017.35 MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Pesin, Y., Zhang, K.: Phase transitions for uniformly expanding maps. J. Stat. Phys. 122(6), 1095–1110 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Pesin, Y., Senti, S., Zhang, K.: Thermodynamics of towers of hyperbolic type. Trans. Amer. Math. Soc. 368(12), 8519–8552 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ruelle, D.: Thermodynamic formalism. Encyclopedia of Mathematics and its Applications, vol. 5. Addison-Wesley, Boston (1978)zbMATHGoogle Scholar
  18. 18.
    Sarig, O.: Lecture Notes on thermodynamics formalism for topological Markov shifts. Penn State, 141pp (2009)Google Scholar
  19. 19.
    Sarig, O.: Thermodynamic formalism for countable Markov shifts. Ergod. Theory Dyn. Syst. 19(6), 1565–1593 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Sarig, O.: Phase transitions for countable Markov shifts. Commun. Math. Phys. 217, 555–577 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Sarig, O.: Subexponential decay of correlations. Invent. Math. 150, 629–653 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Sarig, O.: Existence of Gibbs measure for countable Markov shifts. Proc. Amer. Math. Soc. 131(6), 1751–1758 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Sarig, O.: Bernoulli equilibrium states for surface diffeomorphisms. J. Mod. Dyn. 3, 593–608 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Sarig, O.: Thermodynamic formalism for countable Markov shifts. Proc. Symp. Pure Math. 89, 81–117 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Senti, S., Takahasi, H.: Equilibrium measures for the Hénon map at the first bifurcation. Nonlinearity 26, 1719–1741 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Young, L.-S.: Statistical properties of dynamical systems with some hyperbolicity. Ann. Math. 147(3), 585–650 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Young, L.-S.: Recurrence times and rates of mixing. Israel J. Math. 110, 153–188 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Zelerowicz, A.: Thermodynamics of some non-uniformly hyperbolic attractors. Nonlinearity 30, 2612–2646 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsPennsylvania State UniversityUniversity ParkUSA

Personalised recommendations