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February  2019, 39(2): 959-993. doi: 10.3934/dcds.2019040

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

Xiaojun Huang , , Yuan Lian and  Changrong Zhu

College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

* Corresponding author

Received  March 2018 Revised  August 2018 Published  November 2018

Fund Project: 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).

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.

Keywords: Bowen topological pressure, local pressure of measures, amenable group.
Mathematics Subject Classification: 37A35, 37B40.
Citation: Xiaojun Huang, Yuan Lian, Changrong Zhu. A Billingsley-type theorem for the pressure of an action of an amenable group. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 959-993. doi: 10.3934/dcds.2019040
References:
[1]

R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319.  doi: 10.2307/1994177.  Google Scholar

[2]

P. Billingsley, Ergodic Theory and Information, John Wiley and Sons Inc., New York, 1965.  Google Scholar

[3]

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.  Google Scholar

[4]

R. Bowen, Equilibrium States and the Ergodic Theorey of Anosov Diffeomorphisms, Lecture Notes in Math., vol. 470, Springer-Verlag, 1975.  Google Scholar

[5]

R. Bowen, Hausdorff dimension of quasicircles, Publ. Math. Inst. Hautes Etudes Sci., 50 (1979), 11-25.   Google Scholar

[6]

M. Brin and A. Katok, On local entropy, Lecture Notes in Math., Springer, Berlin, 1007 (1983), 30–38. doi: 10.1007/BFb0061408.  Google Scholar

[7]

C. Carathéodory, Über das lineare mass, Göttingen Nachr., (1914), 406-426.   Google Scholar

[8]

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.  Google Scholar

[9]

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.  Google Scholar

[10]

L. W. Goodwyn, Topological entropy bounds measure-theoretic entropy, Proc. Amer. Math. Soc., 23 (1969), 679-688.  doi: 10.2307/2036610.  Google Scholar

[11]

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.  Google Scholar

[12]

D. Kerr and H. Li, Ergodic Theory: Independence and Dichotomies, Springer, 2016. doi: 10.1007/978-3-319-49847-8.  Google Scholar

[13]

A. N. Kolmogorov, A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces, Dokl. Akad. Nauk SSSR, 119 (1958), 861-864.   Google Scholar

[14]

H. Li, Sofic mean dimension, Adv. Math., 244 (2013), 570-604.  doi: 10.1016/j.aim.2013.05.005.  Google Scholar

[15]

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.  Google Scholar

[16]

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.  Google Scholar

[17]

I. Namioka, Følner's conditions for amenable semi-groups, Math. Scand., 15 (1964), 18-28.  doi: 10.7146/math.scand.a-10723.  Google Scholar

[18]

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.  Google Scholar

[19]

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.  Google Scholar

[20]

Y. Pesin and B. S. Pitskel, Topological pressure and the variational principle for noncompact sets, Functional Anal. Appl., 18 (1984), 50-63, 96.   Google Scholar

[21]

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.  Google Scholar

[22]

D. Ruelle, Thermodynamic Formalism, Vol.5 of Encyclopedia of Mathematics and its Applications. Reading, MA: Addision-Wesley, 1978.  Google Scholar

[23]

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.  Google Scholar

[24]

P. Walters, A variational principle for the pressure of continuous transformations, Amer. J. Math., 97 (1975), 937-971.  doi: 10.2307/2373682.  Google Scholar

show all references

References:
[1]

R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319.  doi: 10.2307/1994177.  Google Scholar

[2]

P. Billingsley, Ergodic Theory and Information, John Wiley and Sons Inc., New York, 1965.  Google Scholar

[3]

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.  Google Scholar

[4]

R. Bowen, Equilibrium States and the Ergodic Theorey of Anosov Diffeomorphisms, Lecture Notes in Math., vol. 470, Springer-Verlag, 1975.  Google Scholar

[5]

R. Bowen, Hausdorff dimension of quasicircles, Publ. Math. Inst. Hautes Etudes Sci., 50 (1979), 11-25.   Google Scholar

[6]

M. Brin and A. Katok, On local entropy, Lecture Notes in Math., Springer, Berlin, 1007 (1983), 30–38. doi: 10.1007/BFb0061408.  Google Scholar

[7]

C. Carathéodory, Über das lineare mass, Göttingen Nachr., (1914), 406-426.   Google Scholar

[8]

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.  Google Scholar

[9]

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.  Google Scholar

[10]

L. W. Goodwyn, Topological entropy bounds measure-theoretic entropy, Proc. Amer. Math. Soc., 23 (1969), 679-688.  doi: 10.2307/2036610.  Google Scholar

[11]

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.  Google Scholar

[12]

D. Kerr and H. Li, Ergodic Theory: Independence and Dichotomies, Springer, 2016. doi: 10.1007/978-3-319-49847-8.  Google Scholar

[13]

A. N. Kolmogorov, A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces, Dokl. Akad. Nauk SSSR, 119 (1958), 861-864.   Google Scholar

[14]

H. Li, Sofic mean dimension, Adv. Math., 244 (2013), 570-604.  doi: 10.1016/j.aim.2013.05.005.  Google Scholar

[15]

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.  Google Scholar

[16]

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.  Google Scholar

[17]

I. Namioka, Følner's conditions for amenable semi-groups, Math. Scand., 15 (1964), 18-28.  doi: 10.7146/math.scand.a-10723.  Google Scholar

[18]

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.  Google Scholar

[19]

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.  Google Scholar

[20]

Y. Pesin and B. S. Pitskel, Topological pressure and the variational principle for noncompact sets, Functional Anal. Appl., 18 (1984), 50-63, 96.   Google Scholar

[21]

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.  Google Scholar

[22]

D. Ruelle, Thermodynamic Formalism, Vol.5 of Encyclopedia of Mathematics and its Applications. Reading, MA: Addision-Wesley, 1978.  Google Scholar

[23]

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.  Google Scholar

[24]

P. Walters, A variational principle for the pressure of continuous transformations, Amer. J. Math., 97 (1975), 937-971.  doi: 10.2307/2373682.  Google Scholar

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