Advanced Search
Article Contents
Article Contents

A Billingsley-type theorem for the pressure of an action of an amenable group

  • * Corresponding author

    * Corresponding author 

The research was supported by NSF of China (No. 11671057, No. 11471318, No. 11671058) and the Fundamental Research Funds for the Central Universities (No. 2018CDQYST0023)

Abstract / Introduction Full Text(HTML) Related Papers Cited by
  • This paper extends the definition of Bowen topological entropy of subsets to Pesin-Pitskel topological pressure for the continuous action of amenable groups on a compact metric space. We introduce the local measure theoretic pressure of subsets and investigate the relation between local measure theoretic pressure of Borel probability measures and Pesin-Pitskel topological pressure on an arbitrary subset of a compact metric space.

    Mathematics Subject Classification: 37A35, 37B40.


    \begin{equation} \\ \end{equation}
  • 加载中
  •   R. L. Adler , A. G. Konheim  and  M. H. McAndrew , Topological entropy, Trans. Amer. Math. Soc., 114 (1965) , 309-319.  doi: 10.2307/1994177.
      P. Billingsley, Ergodic Theory and Information, John Wiley and Sons Inc., New York, 1965.
      A. Bis , An analogue of the variational principle for group and pseudogroup actions, Ann. Inst. Fourier (Grenoble), 63 (2013) , 839-863.  doi: 10.5802/aif.2778.
      R. Bowen, Equilibrium States and the Ergodic Theorey of Anosov Diffeomorphisms, Lecture Notes in Math., vol. 470, Springer-Verlag, 1975.
      R. Bowen , Hausdorff dimension of quasicircles, Publ. Math. Inst. Hautes Etudes Sci., 50 (1979) , 11-25. 
      M. Brin and A. Katok, On local entropy, Lecture Notes in Math., Springer, Berlin, 1007 (1983), 30–38. doi: 10.1007/BFb0061408.
      C. Carathéodory , Über das lineare mass, Göttingen Nachr., (1914) , 406-426. 
      D. Feng  and  W. Huang , Variational principles for topological entropies of subsets, J. Funct. Anal., 263 (2012) , 2228-2254.  doi: 10.1016/j.jfa.2012.07.010.
      T. N. T. Goodman , Relating topological entropy and measure entropy, Bull. London Math. Soc., 3 (1971) , 176-180.  doi: 10.1112/blms/3.2.176.
      L. W. Goodwyn , Topological entropy bounds measure-theoretic entropy, Proc. Amer. Math. Soc., 23 (1969) , 679-688.  doi: 10.2307/2036610.
      X. Huang , J. Liu  and  C. Zhu , The Bowen topological entropy of subsets for amenable group actions, Discrete Contin. Dyn. Syst., 38 (2018) , 4467-4482.  doi: 10.3934/dcds.2018195.
      D. Kerr and H. Li, Ergodic Theory: Independence and Dichotomies, Springer, 2016. doi: 10.1007/978-3-319-49847-8.
      A. N. Kolmogorov , A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces, Dokl. Akad. Nauk SSSR, 119 (1958) , 861-864. 
      H. Li , Sofic mean dimension, Adv. Math., 244 (2013) , 570-604.  doi: 10.1016/j.aim.2013.05.005.
      B. Liang  and  K. Yan , Topological pressure for sub-additive potentials of amenable group actions, J. Funct. Anal., 262 (2012) , 584-601.  doi: 10.1016/j.jfa.2011.09.020.
      J. Ma and Z. Wen, A Billingsley type theorem for Bowen entropy, C. R. Math. Acad. Sci., Paris, 346 (2008), 503–507. doi: 10.1016/j.crma.2008.03.010.
      I. Namioka , Følner's conditions for amenable semi-groups, Math. Scand., 15 (1964) , 18-28.  doi: 10.7146/math.scand.a-10723.
      D. S. Ornstein  and  B. Weiss , Entropy and isomorphism theorems for actions of amenable groups, J. Analyse Math., 48 (1987) , 1-141.  doi: 10.1007/BF02790325.
      Y. B. Pesin, Dimension Theory in Dynamical Systems Contemporary Views and Applications, Chicago lecture in Mathematics. University of Chicago Press, Chicago, IL, 1997. doi: 10.7208/chicago/9780226662237.001.0001.
      Y. Pesin  and  B. S. Pitskel , Topological pressure and the variational principle for noncompact sets, Functional Anal. Appl., 18 (1984) , 50-63, 96. 
      D. Ruelle , Statistical mechanics on compact set with Zv action satisfying expansiveness and specification, Trans. Amer. Math. Soc., 187 (1973) , 237-251.  doi: 10.2307/1996437.
      D. Ruelle, Thermodynamic Formalism, Vol.5 of Encyclopedia of Mathematics and its Applications. Reading, MA: Addision-Wesley, 1978.
      X. Tang , W. Cheng  and  Y. Zhao , Variational principle for topological pressures on subsets, J. Math. Anal. Appl., 424 (2015) , 1272-1285.  doi: 10.1016/j.jmaa.2014.11.066.
      P. Walters , A variational principle for the pressure of continuous transformations, Amer. J. Math., 97 (1975) , 937-971.  doi: 10.2307/2373682.
  • 加载中

Article Metrics

HTML views(1615) PDF downloads(267) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint