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May  2020, 40(5): 2687-2703. doi: 10.3934/dcds.2020146

A variational principle of topological pressure on subsets for amenable group actions

Xiaojun Huang , , Zhiqiang Li and  Yunhua Zhou

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

* Corresponding authors

Received  May 2019 Published  March 2020

In this paper, we establish a variational principle for topological pressure on compact subsets in the context of amenable group actions. To be precise, for a countable amenable group action on a compact metric space, say $ G\curvearrowright X $, for any potential $ f\in C(X) $, we define and study topological pressure on an arbitrary subset and measure theoretic pressure for any Borel probability measure on $ X $ (not necessarily invariant); moreover, we prove a variational principle for this topological pressure on a given nonempty compact subset $ K\subseteq X $.

Keywords: Amenable groups, Pesin-Pitskel topological pressure, local measure theoretic pressure, variational principle.
Mathematics Subject Classification: 37A35, 37B40.
Citation: Xiaojun Huang, Zhiqiang Li, Yunhua Zhou. A variational principle of topological pressure on subsets for amenable group actions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (5) : 2687-2703. doi: 10.3934/dcds.2020146
References:
[1]

R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319.  doi: 10.1090/S0002-9947-1965-0175106-9.  Google Scholar

[2]

L. Bowen, Sofic entropy and amenable groups, Ergodic Theory Dynam. Systems, 32 (2012), 427-466.  doi: 10.1017/S0143385711000253.  Google Scholar

[3]

L. Bowen and A. Nevo, Pointwise ergodic theorems beyond amenable groups, Ergodic Theory Dynam. Systems, 33 (2013), 777-820.  doi: 10.1017/S0143385712000041.  Google Scholar

[4]

R. Bowen, Equilibrium States and the Ergodic Theorey of Anosov Diffeomorphisms, Lecture Notes in Mathematics, 470. Springer-Verlag, Berlin, 2008.  Google Scholar

[5]

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

[6]

D.-J. 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

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M. Hochman, Return times, recurrence densities and entropy for actions of some discrete amenable groups, J. Anal. Math., 100 (2006), 1-51.  doi: 10.1007/BF02916754.  Google Scholar

[8]

X. J. Huang, J. S. Liu and C. R. Zhu, The Katok's entropy formula for discrete amenable group actions, Discrete Contin. Dyn. Syst., 38 (2018), 4467-4482.  doi: 10.3934/dcds.2018195.  Google Scholar

[9]

X. J. Huang, Y. Lian and C. R. Zhu, A Billingsley-type theorem for the pressure of an amenable group, Discrete Contin. Dyn. Syst., 39 (2018), 959-993.  doi: 10.3934/dcds.2019040.  Google Scholar

[10]

W. Huang, X. D. Ye and G. H. Zhang, Local entropy theory for a countable discrete amenable group action, J. Funct. Anal., 261 (2011), 1028-1082.  doi: 10.1016/j.jfa.2011.04.014.  Google Scholar

[11]

D. Kerr and H. F. Li, Entropy and variational principle for actions of sofic groups, Invent. Math., 186 (2011), 501-558.  doi: 10.1007/s00222-011-0324-9.  Google Scholar

[12]

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

[13]

B. B. Liang and K. S. 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

[14]

E. Lindenstrauss, Pointwise theorems for amenable groups, Invent. Math., 146 (2001), 259-295.  doi: 10.1007/s002220100162.  Google Scholar

[15]

P. Mattila, Geometry of Sets and Measure in Euclideans Spaces: Fractals and Rectifiability, Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511623813.  Google Scholar

[16]

M. A. Misiurewicz, A short proof of the variational principle for a $\mathbb{Z}^{n}_{+}$-action on a compact space, Bull. Acad. Pol. Sci. Ser. Sci. Math. Astron. Phys., 24 (1976), 1069-1075.   Google Scholar

[17]

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

[18] 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
[19]

Ya. B. Pesin and B. S. Pitskel', Topological pressure and the variational principle for non-compact sets, Funktsional. Anal. i Prilozhen., 18 (1984), 50–63, 96.  Google Scholar

[20]

D. Ruelle, Statistical mechanics on compact set with $\mathbb{Z}^{v}$ action satisfying expansiveness and specification, Trans. Amer. Math. Soc., 187 (1973), 237-251.  doi: 10.2307/1996437.  Google Scholar

[21]

D. Ruelle, Thermodynamic Formalism, Encyclopedia of Mathematics and its Applications, 5. Addison-Wesley Publishing Co., Reading, Mass., 1978.  Google Scholar

[22]

D. J. Rudolph and B. Weiss, Entropy and mixing for amenable group actions, Ann. of Math., 151 (2000), 1119-1150.  doi: 10.2307/121130.  Google Scholar

[23]

X. J. Tang, W.-C. 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

[25]

C. W. Wang and E. Chen, Variational principles for BS dimension of subsets, Dyn. Syst., 27 (2012), 359-385.  doi: 10.1080/14689367.2012.702419.  Google Scholar

[26]

Y. H. Zhou, Tail variational principle for a countable discrete amenable group action, J. Math. Anal. Appl., 433 (2016), 1513-1530.  doi: 10.1016/j.jmaa.2015.08.058.  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: 10.1090/S0002-9947-1965-0175106-9.  Google Scholar

[2]

L. Bowen, Sofic entropy and amenable groups, Ergodic Theory Dynam. Systems, 32 (2012), 427-466.  doi: 10.1017/S0143385711000253.  Google Scholar

[3]

L. Bowen and A. Nevo, Pointwise ergodic theorems beyond amenable groups, Ergodic Theory Dynam. Systems, 33 (2013), 777-820.  doi: 10.1017/S0143385712000041.  Google Scholar

[4]

R. Bowen, Equilibrium States and the Ergodic Theorey of Anosov Diffeomorphisms, Lecture Notes in Mathematics, 470. Springer-Verlag, Berlin, 2008.  Google Scholar

[5]

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

[6]

D.-J. 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

[7]

M. Hochman, Return times, recurrence densities and entropy for actions of some discrete amenable groups, J. Anal. Math., 100 (2006), 1-51.  doi: 10.1007/BF02916754.  Google Scholar

[8]

X. J. Huang, J. S. Liu and C. R. Zhu, The Katok's entropy formula for discrete amenable group actions, Discrete Contin. Dyn. Syst., 38 (2018), 4467-4482.  doi: 10.3934/dcds.2018195.  Google Scholar

[9]

X. J. Huang, Y. Lian and C. R. Zhu, A Billingsley-type theorem for the pressure of an amenable group, Discrete Contin. Dyn. Syst., 39 (2018), 959-993.  doi: 10.3934/dcds.2019040.  Google Scholar

[10]

W. Huang, X. D. Ye and G. H. Zhang, Local entropy theory for a countable discrete amenable group action, J. Funct. Anal., 261 (2011), 1028-1082.  doi: 10.1016/j.jfa.2011.04.014.  Google Scholar

[11]

D. Kerr and H. F. Li, Entropy and variational principle for actions of sofic groups, Invent. Math., 186 (2011), 501-558.  doi: 10.1007/s00222-011-0324-9.  Google Scholar

[12]

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

[13]

B. B. Liang and K. S. 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

[14]

E. Lindenstrauss, Pointwise theorems for amenable groups, Invent. Math., 146 (2001), 259-295.  doi: 10.1007/s002220100162.  Google Scholar

[15]

P. Mattila, Geometry of Sets and Measure in Euclideans Spaces: Fractals and Rectifiability, Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511623813.  Google Scholar

[16]

M. A. Misiurewicz, A short proof of the variational principle for a $\mathbb{Z}^{n}_{+}$-action on a compact space, Bull. Acad. Pol. Sci. Ser. Sci. Math. Astron. Phys., 24 (1976), 1069-1075.   Google Scholar

[17]

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

[18] 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
[19]

Ya. B. Pesin and B. S. Pitskel', Topological pressure and the variational principle for non-compact sets, Funktsional. Anal. i Prilozhen., 18 (1984), 50–63, 96.  Google Scholar

[20]

D. Ruelle, Statistical mechanics on compact set with $\mathbb{Z}^{v}$ action satisfying expansiveness and specification, Trans. Amer. Math. Soc., 187 (1973), 237-251.  doi: 10.2307/1996437.  Google Scholar

[21]

D. Ruelle, Thermodynamic Formalism, Encyclopedia of Mathematics and its Applications, 5. Addison-Wesley Publishing Co., Reading, Mass., 1978.  Google Scholar

[22]

D. J. Rudolph and B. Weiss, Entropy and mixing for amenable group actions, Ann. of Math., 151 (2000), 1119-1150.  doi: 10.2307/121130.  Google Scholar

[23]

X. J. Tang, W.-C. 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

[25]

C. W. Wang and E. Chen, Variational principles for BS dimension of subsets, Dyn. Syst., 27 (2012), 359-385.  doi: 10.1080/14689367.2012.702419.  Google Scholar

[26]

Y. H. Zhou, Tail variational principle for a countable discrete amenable group action, J. Math. Anal. Appl., 433 (2016), 1513-1530.  doi: 10.1016/j.jmaa.2015.08.058.  Google Scholar

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