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

Xiaojun Huang; Zhiqiang Li; Yunhua Zhou

doi:10.3934/dcds.2020146

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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 and Continuous Dynamical Systems, 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.
[2]	L. Bowen, Sofic entropy and amenable groups, Ergodic Theory Dynam. Systems, 32 (2012), 427-466. doi: 10.1017/S0143385711000253.
[3]	L. Bowen and A. Nevo, Pointwise ergodic theorems beyond amenable groups, Ergodic Theory Dynam. Systems, 33 (2013), 777-820. doi: 10.1017/S0143385712000041.
[4]	R. Bowen, Equilibrium States and the Ergodic Theorey of Anosov Diffeomorphisms, Lecture Notes in Mathematics, 470. Springer-Verlag, Berlin, 2008.
[5]	R. Bowen, Hausdorff dimension of quasicircles, Inst. Hautes Études Sci. Publ. Math., (1979), 11–25.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[14]	E. Lindenstrauss, Pointwise theorems for amenable groups, Invent. Math., 146 (2001), 259-295. doi: 10.1007/s002220100162.
[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.
[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.
[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.
[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.
[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.
[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.
[21]	D. Ruelle, Thermodynamic Formalism, Encyclopedia of Mathematics and its Applications, 5. Addison-Wesley Publishing Co., Reading, Mass., 1978.
[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.
[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.
[24]	P. Walters, A variational principle for the pressure of continuous transformations, Amer. J. Math., 97 (1975), 937-971. doi: 10.2307/2373682.
[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.
[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.

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.
[2]	L. Bowen, Sofic entropy and amenable groups, Ergodic Theory Dynam. Systems, 32 (2012), 427-466. doi: 10.1017/S0143385711000253.
[3]	L. Bowen and A. Nevo, Pointwise ergodic theorems beyond amenable groups, Ergodic Theory Dynam. Systems, 33 (2013), 777-820. doi: 10.1017/S0143385712000041.
[4]	R. Bowen, Equilibrium States and the Ergodic Theorey of Anosov Diffeomorphisms, Lecture Notes in Mathematics, 470. Springer-Verlag, Berlin, 2008.
[5]	R. Bowen, Hausdorff dimension of quasicircles, Inst. Hautes Études Sci. Publ. Math., (1979), 11–25.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[14]	E. Lindenstrauss, Pointwise theorems for amenable groups, Invent. Math., 146 (2001), 259-295. doi: 10.1007/s002220100162.
[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.
[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.
[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.
[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.
[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.
[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.
[21]	D. Ruelle, Thermodynamic Formalism, Encyclopedia of Mathematics and its Applications, 5. Addison-Wesley Publishing Co., Reading, Mass., 1978.
[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.
[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.
[24]	P. Walters, A variational principle for the pressure of continuous transformations, Amer. J. Math., 97 (1975), 937-971. doi: 10.2307/2373682.
[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.
[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.

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