Uniqueness and Continuity of Equilibrium States for DA-Diffeomorphisms

• Abbas Fakhari ^[1] ; Mohammad Khalili Garakani ^[1]
  1. [1] Shahid Beheshti University
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 21, Nº 4, 2022
• Idioma: inglés
• Enlaces
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• Resumen

  • In this paper we focused our study on Derived from Anosov diffeomorphisms (DAdiffeomorphisms) of T3, that is, an absolutely partially hyperbolic diffeomorphisms on T3 which are homotopic and semi-conjugate to an Anosov linear automorphism of T3. In this class of dynamics, the existence of ergodic equilibrium states is known, we study the uniqueness and continuity properties of such equilibrium states for lifted potential functions.

• Referencias bibliográficas
  • 1. Alves, J., Viana, M.: Statistical stability for robust classes of maps with non-uniform expansion. Ergodic Theory Dyn. Syst. 22, 1–32 (2002)
  • 2. Alves, J.: Strong statistical stability of non-uniformly expanding maps. Nonlinearity 17(4), 1193–1215 (2004)
  • 3. Alves, J., Ramos, V., Siqueira, J.: Equilibrium stability for non-uniformly hyperbolic systems. Ergodic Theory Dyn. Syst. 39(10), 2619–2642...
  • 4. Bowen, R.: Equilibrium States and Ergodic Theory of Anosov Diffeomorphisms. Lecture notes in mathematics, Springer-Verlag, 470 (1975)
  • 5. Brin, M., Burago, D., Ivanov, S.: Dynamical coherence of partially hyperbolic diffeomorphisms of the 3-torus. J. Mod. Dyn. 3, 1–11 (2009)
  • 6. Burago, D., Ivanov, S.: Partially hyperbolic diffeomorphisms of 3-manifolds with abelian fundamental groups. J. Mod. Dyn. 2, 541–580 (2008)
  • 7. Buzzi, J., Fisher, T., Sambarino, M., Vasquez, C.: Maximal entropy measure for certain partially hyperbolic, derived from Anosov systems....
  • 8. Cowieson, W., Young, L.S.: SRB measures as zero-noise limit. Ergodic Theory Dyn. Syst. 25(4), 1115–1138 (2005)
  • 9. Climenhaga, V., Fisher, T., Thompson, J.: Equilibrium states for Mañé diffeomorphisms. Ergodic Theory Dyn. Syst. 39(9), 2433–2455 (2019)
  • 10. Crisostomo, J., Tahzibi, A.: Equilibrium states for partially hyperbolic diffeomorphisms with hyperbolic linear part. Nonlinearity 32(2),...
  • 11. Diaz, L., Fisher, T., Pacifico, M., Vieitez, J.: Entropy expensiveness for partially hyperbolic diffeomorphisms. Discrete Contin. Dyn....
  • 12. Franks, J.: Anosov diffeomorphisms on tori. Trans. Am. Math. Soc. 145, 117–124 (1969)
  • 13. Freitas, J.M.: Continuity of SRB measure and entropy for Benedicks-Carleson quadratic maps. Nonlinearity 18, 831–854 (2005)
  • 14. Freitas, J.M., Todd, M.: The statistical stability of equilibrium states for interval maps. Nonlinearity 22, 259–281 (2009)
  • 15. Gan, S., Shi, Y.: Rigidity of center Lyapunov exponents and su-integrability. Comment. Math. Helv. 95, 569–592 (2020)
  • 16. Hertz, F.R., Hertz, M.R., Ures, R.: A non-dynamical coherent example in T3. Ann. I. H. Poincaré AN 33(4), 1023–1032 (2016)
  • 17. Hua, Y., Saghin, R., Xia, Z.: Topological entropy and partially hyperbolic diffeomorphisms. Ergodic Theory Dyn. Syst. 28, 843–862 (2008)
  • 18. Ledrappier, F., Walters, L.: A relativised variational principle for continuous transformation. J. London Math. Soc. s2–16(3), 568–576...
  • 19. Ledrappier, F., Young, L.S.: The metric entropy of diffeomorphisms: Part II: Relations between entropy, exponents and dimension. Ann....
  • 20. Mañé, R.: Contributions to the stability conjecture. Topology 17, 383–396 (1978)
  • 21. Misiurewicz, M.: Diffeomorphism without any measures with maximal entropy. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 21,...
  • 22. Ponce, G., Tahzibi, A., Varão, R.: Minimal yet measurable foliations. J. Mod. Dyn. 8, 93–107 (2014)
  • 23. Pilyugin, S.Y., Sakai, K.: Shadowing and Hyperbolicity. Lectures Notes in Mathematics, Springer, (2017)
  • 24. Raith, P.: Continuity of the measure of maximal entropy for unimodal maps on the interval. Qual Theory Dyn. Syst. 4(53), 67–76 (2003)
  • 25. Ramos, V., Viana, M.: Equilibrium states for hyperbolic potentials. Nonlinearity 30(2), 825–847 (2017)
  • 26. Ures, R.: Intrinsic ergodicity of partially hyperbolic diffeomorphisms with hyperbolic linear part. Proc. Am. Math. Soc. 140(6), 1973–1985...
  • 27. Varandas, P., Viana, M.: Existence, uniqueness and stability of equilibrium states for non-uniformly expanding maps. Ann. I. H. Poincaré...
  • 28. Varão, R.: Center foliation: absolute continuity, disintegration and rigidity. Ergodic Theory Dyn. Syst. 36(1), 256–275 (2016)
  • 29. Veconi, D.: Equilibrium states of almost Anosov diffeomorphisms. Discrete Contin. Dyn. Syst. 40(2), 767–780 (2020)
  • 30. Viana, M., Yang, J.: Measure theoretical properties of center foliation. Mod. Theory Dyn. Syst. 692, 291–320 (2017)