Variational Principle of Topological Pressure of Free Semigroup Actions for Subsets

• Qian Xiao ^[1] ; Dongkui Ma
  1. [1] South China University of Technology

     South China University of Technology

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 21, Nº 4, 2022
• Idioma: inglés
• Enlaces
  • Texto completo (pdf)
• Resumen

  • In this paper, we investigate the relation between topological pressure of free semigroup actions for non-compact sets proposed by Xiao and Ma (J Dynam Differ Equ, 2021) and measure-theoretic pressure of Borel probability measure. For any Borel probability measure, this paper defines lower and upper measure-theoretic pressures.

    Moreover, we give a lower and an upper estimations of the topological pressure of free semigroup actions by local measure-theoretic pressure. In addition, we also show that the topological pressure on a non-empty compact subset K defined in Xiao and Ma (J Dynam Differ Equ, 2021) equals to the supremum of the lower measure-theoretic pressure taken over all probability measures supported on K.

• Referencias bibliográficas
  • 1. Adler, R.L., Konheim, A.G., McAndrew, M.H.: Topological entropy. Trans. Amer. Math. Soc. 114, 309–319 (1965)
  • 2. Ban, J., Cao, Y., Hu, H.: The dimensions of a non-conformal repeller and an average conformal repeller. Trans. Amer. Math. Soc. 362(2),...
  • 3. Bi´s, A.: Entropies of a semigroup of maps. Discrete Contin. Dyn. Systs. Series A. 11, 639–648 (2004)
  • 4. Bowen, R.: Topological entropy for noncompact sets. Trans. Amer. Math. Soc. 184, 125–136 (1973)
  • 5. Brin, M., Katok, A.: On local entropy, In: Geometric dynamics (Rio de Janeiro, 1981), Vol. 1007 of Lecture Notes in Math., Berlin: Springer,...
  • 6. Bufetov, A.: Topological entropy of free semigroup actions and skew-product transformations. J. Dynam. Control Systems 5(1), 137–143 (1999)
  • 7. Carvalho, M., Rodrigues, F., Varandas, P.: A variational principle for free semigroup actions. Adv. Math. 334, 450–487 (2018)
  • 8. Climenhaga, V.: Thermodynamic formalism and multifractal analysis for general topological dynamical systems, Thesis (Ph.D.)-The Pennsylvania...
  • 9. Dinaburg, E.I.: The relation between topological entropy and metric entropy. Sov. Math. 11, 13–16 (1970)
  • 10. Feng, D., Huang, W.: Variational principles for topological entropies of subsets. J. Funct. Anal. 263(8), 2228–2254 (2012)
  • 11. Federer, H.: Geometric Measure Theory. Springer-Verlag, New York (1969)
  • 12. Ghys, É., Langevin, R., Walczak, P.: Entropie géométrique des feuilletages. Acta Math. 160, 105–142 (1988)
  • 13. Goodman, T.N.T.: Relating topological entropy and measure entropy. Bull. Lond. Math. Soc. 3, 176– 180 (1971)
  • 14. Goodwyn, L.W.: Topological entropy bounds measure-throretic entropy. Proc. Amer. Math. Soc. 23, 679–688 (1969)
  • 15. Ju, Y., Ma, D., Wang, Y.: Topological entropy of free semigroup actions for noncompact sets. Discrete Contin. Dyn. Syst. 39(2), 995–1017...
  • 16. Kolmogorov, A.N.: A new metric invariant of transient dynamical systems and automorphisms of Lebesgue spaces. Dokl. Akad. Sci. SSSR 119,...
  • 17. Lin, X., Ma, D., Wang, Y.: On the measure-theoretic entropy and topological pressure of free semigroup actions. Ergodic Theory & Dynam....
  • 18. Ma, D., Wu, M.: Topological pressure and topological entropy of a semigroup of maps. Discrete Contin. Dyn. Systs. 31, 545–557 (2011)
  • 19. Ma, J., Wen, Z.: A Billingsley type theorem for Bowen entropy. C. R. Math. Acad. Sci. Paris 346, 503–507 (2008)
  • 20. Mattlia, P.: Geometry of Sets and Measures in Euclidean Space, Cambridge University Press, (1995)
  • 21. Pesin, Y.: Dimension Theory in Dynamical Systems. The university of Chicago Press, Chicago (1997)
  • 22. Pesin, Y., Pitskel, B.S.: Topological pressure and the variational principle for noncompact sets. Functional Anal. and Its Applications...
  • 23. Tang, X., Cheng, W., Zhao, Y.: Variational principle for topological pressure on subsets. J. Math. Anal. Appl. 424(2), 1272–1285 (2015)
  • 24. Wang, Y., Ma, D.: On the topological entropy of a semigroup of continuous maps. J. Math. Anal. Appl. 427(2), 1084–1100 (2015)
  • 25. Xiao, Q., Ma, D.: Topological pressure of free semigroup actions for non-compact sets and Bowen’s equation, II. J. Dynam. Differ. Equ....
  • 26. Zhong, X., Chen, Z.: Variational principle for topological pressure on subsets of free semigroup actions. Acta Math. Sin. (Engl. Ser.),...