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

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.

Citation: Xiaojun Huang, Yuan Lian, Changrong Zhu. A Billingsley-type theorem for the pressure of an action of an amenable group. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 959-993. doi: 10.3934/dcds.2019040
References:
[1]

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

[2]

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

[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.

[4]

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

[5]

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

[6]

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

[7]

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

[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.

[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.

[10]

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

[11]

X. HuangJ. 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.

[12]

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

[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. 

[14]

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

[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.

[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.

[17]

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

[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.

[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.

[20]

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

[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.

[22]

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

[23]

X. TangW. 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.

[24]

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

show all references

References:
[1]

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

[2]

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

[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.

[4]

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

[5]

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

[6]

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

[7]

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

[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.

[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.

[10]

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

[11]

X. HuangJ. 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.

[12]

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

[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. 

[14]

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

[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.

[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.

[17]

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

[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.

[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.

[20]

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

[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.

[22]

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

[23]

X. TangW. 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.

[24]

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

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