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A variational principle of topological pressure on subsets for amenable group actions
College of Mathematics and Statistics, Chongqing University, Chongqing, China 401331 |
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 $.
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. |
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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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