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April  2018, 38(4): 1657-1667. doi: 10.3934/dcds.2018068

Periodic measures are dense in invariant measures for residually finite amenable group actions with specification

Xiankun Ren 1,2,

1. 

School of Mathematical Sciences, Peking University, Beijing 100871, China
2. 

Department of Mathematics, SUNY at Buffalo, Buffalo, NY 14260-2900, USA

Received  January 2016 Revised  October 2017 Published  January 2018

Fund Project: The author is supported by NNSFC grant No.11471344.

We prove that for actions of a discrete countable residuallyfinite amenable group on a compact metric space with specification property, periodic measures are dense in theset of invariant measures. We also prove that certain expansiveactions of a countable discrete group by automorphisms of compact abelian groups have specification property.

Keywords: Residually finite amenable group action, periodic measure, specification property.
Mathematics Subject Classification: Primary: 37B40, 37A35, 22D25.
Citation: Xiankun Ren. Periodic measures are dense in invariant measures for residually finite amenable group actions with specification. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1657-1667. doi: 10.3934/dcds.2018068
References:
[1]

M. Abért, A. Jaikin-Zapirain and N. Nikolay, The rank gradient from a combinatorial viewpoint, Groups Geom. Dyn., 5 (2011), 213-230.  doi: 10.4171/GGD/124.  Google Scholar

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F. Abdenur, C. Bonatti and S. Crovisier, Nonuniform hyperbolicity for C$^{1}$-generic diffeomorphisms, Israel J. Math., 183 (2011), 1-60.  doi: 10.1007/s11856-011-0041-5.  Google Scholar

[3]

R. Bowen, Periodic points and measures for Axiom A diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397.   Google Scholar

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B. F. Bryant, On expansive homeomorphisms, Pacific J. Math., 10 (1960), 1163-1167.  doi: 10.2140/pjm.1960.10.1163.  Google Scholar

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T. Ceccherini-Silberstein and M. Coornaert, Cellular Automata and Groups, Springer Monographs in Mathematics, Springer-Verlag, New York, Berlin, 2010. doi: 978-3-642-14033-4.  Google Scholar

[6]

N. P. Chung and H. Li, Homoclinic group, IE group, and expansive algebraic actions, Invent. Math., 199 (2015), 805-858.  doi: 10.1007/s00222-014-0524-1.  Google Scholar

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M. Coornaert, Topological Dimension and Dynamical Systems, Universitext, Springer, Cham, 2015. doi: 978-3-319-19793-7.  Google Scholar

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C. Deninger and K. Schmidt, Expansive algebraic of discrete residually finite amenable groups and their entropy, Ergod. Th. Dynamical Sys., 27 (2007), 769-786.  doi: 10.1017/S0143385706000939.  Google Scholar

[9]

M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces, Lecture Notes in Mathematics, Vol. 527. Springer-Verlag, Berlin-New York, 1976. Google Scholar

[10]

M. Einsiedler and T. Ward, Ergodic Theory with a View Towards Number Theory, Graduate Texts in Mathematics, vol. 259, Springer-Verlag London Ltd., London, 2011. doi: 978-0-85729-020-5.  Google Scholar

[11]

M. Hirayama, Periodic probability measures are dense in the set of invariant measures, Dist. Cont. Dyn. Sys., 9 (2003), 1185-1192.  doi: 10.3934/dcds.2003.9.1185.  Google Scholar

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

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C. Liang, G. Liu and W. Sun, Approxiamation properties on invariant measures and Oseledec splitting in non-uniformly hyperbolic systems, Trans. Amer. Math. Soc., 361 (2009), 1543-1579.   Google Scholar

[14]

E. Lindenstrauss, Pointwise theorems for amenable groups, Electronic Research Announcements of the American Mathematical Society, 5 (1999), 82-90.  doi: 10.1090/S1079-6762-99-00065-7.  Google Scholar

[15]

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

[16]

K. Oliverira and X. Tian, Non-uniform hyperbolicity and non-uniform specification, Trans. Amer. Math. Soc., 365 (2013), 4371-4392.  doi: 10.1090/S0002-9947-2013-05819-9.  Google Scholar

[17]

D. Ornstein and B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Anal. Math., 48 (1987), 1-141.  doi: 10.1007/BF02790325.  Google Scholar

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C.-E. Pfister and W. Sullivan, On the topological entropy of saturated sets, Ergod. Th. Dynamical Sys., 27 (2007), 929-956.  doi: 10.1017/S0143385706000824.  Google Scholar

[19]

D. Ruelle, Statisticle mechanics on a compact set with $Z^{ν}$ actions satisfying expansiveness and specification, Trans. Amer. Math. Soc., 187 (1973), 237-251.   Google Scholar

[20]

K. Sigmund, Generic properties of invariant measures for Axiom A-diffeomorphisms, Invent. Math., 11 (1970), 99-109.  doi: 10.1007/BF01404606.  Google Scholar

[21]

K. Sigmund, On dynamical systems with specification property, Trans, Amer, Math. Soc., 190 (1974), 285-299.  doi: 10.1090/S0002-9947-1974-0352411-X.  Google Scholar

[22]

D. Tompson, Irregular sets, the beta-transformation and the almost specification property, Trans. Am. Math. Soc., 364 (2012), 5395-5414.  doi: 10.1090/S0002-9947-2012-05540-1.  Google Scholar

[23]

T. Ward and Q. Zhang, The Abramov-Rokhlin entropy addition formular for amenable group actions, Monatsh. Math., 114 (1992), 317-329.  doi: 10.1007/BF01299386.  Google Scholar

[24]

B. Weiss, Monotileable amenable groups, Topology, Ergodic Theory, Real Algebraic Geometry, in: Amer. Math. Soc. Transl. Ser. 2, vol. 202, Amer. Math. Soc., Providence, RI, (2001), 257-262.   Google Scholar

[25]

D. Zheng, E. Chen and J. Yang, On large deviations for amenable group actions, Discrete Contin. Dyn. Syst., 36 (2016), 7191-7206, arXiv:1507.05130. doi: 10.3934/dcds.2016113.  Google Scholar

show all references

References:
[1]

M. Abért, A. Jaikin-Zapirain and N. Nikolay, The rank gradient from a combinatorial viewpoint, Groups Geom. Dyn., 5 (2011), 213-230.  doi: 10.4171/GGD/124.  Google Scholar

[2]

F. Abdenur, C. Bonatti and S. Crovisier, Nonuniform hyperbolicity for C$^{1}$-generic diffeomorphisms, Israel J. Math., 183 (2011), 1-60.  doi: 10.1007/s11856-011-0041-5.  Google Scholar

[3]

R. Bowen, Periodic points and measures for Axiom A diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397.   Google Scholar

[4]

B. F. Bryant, On expansive homeomorphisms, Pacific J. Math., 10 (1960), 1163-1167.  doi: 10.2140/pjm.1960.10.1163.  Google Scholar

[5]

T. Ceccherini-Silberstein and M. Coornaert, Cellular Automata and Groups, Springer Monographs in Mathematics, Springer-Verlag, New York, Berlin, 2010. doi: 978-3-642-14033-4.  Google Scholar

[6]

N. P. Chung and H. Li, Homoclinic group, IE group, and expansive algebraic actions, Invent. Math., 199 (2015), 805-858.  doi: 10.1007/s00222-014-0524-1.  Google Scholar

[7]

M. Coornaert, Topological Dimension and Dynamical Systems, Universitext, Springer, Cham, 2015. doi: 978-3-319-19793-7.  Google Scholar

[8]

C. Deninger and K. Schmidt, Expansive algebraic of discrete residually finite amenable groups and their entropy, Ergod. Th. Dynamical Sys., 27 (2007), 769-786.  doi: 10.1017/S0143385706000939.  Google Scholar

[9]

M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces, Lecture Notes in Mathematics, Vol. 527. Springer-Verlag, Berlin-New York, 1976. Google Scholar

[10]

M. Einsiedler and T. Ward, Ergodic Theory with a View Towards Number Theory, Graduate Texts in Mathematics, vol. 259, Springer-Verlag London Ltd., London, 2011. doi: 978-0-85729-020-5.  Google Scholar

[11]

M. Hirayama, Periodic probability measures are dense in the set of invariant measures, Dist. Cont. Dyn. Sys., 9 (2003), 1185-1192.  doi: 10.3934/dcds.2003.9.1185.  Google Scholar

[12]

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

[13]

C. Liang, G. Liu and W. Sun, Approxiamation properties on invariant measures and Oseledec splitting in non-uniformly hyperbolic systems, Trans. Amer. Math. Soc., 361 (2009), 1543-1579.   Google Scholar

[14]

E. Lindenstrauss, Pointwise theorems for amenable groups, Electronic Research Announcements of the American Mathematical Society, 5 (1999), 82-90.  doi: 10.1090/S1079-6762-99-00065-7.  Google Scholar

[15]

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

[16]

K. Oliverira and X. Tian, Non-uniform hyperbolicity and non-uniform specification, Trans. Amer. Math. Soc., 365 (2013), 4371-4392.  doi: 10.1090/S0002-9947-2013-05819-9.  Google Scholar

[17]

D. Ornstein and B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Anal. Math., 48 (1987), 1-141.  doi: 10.1007/BF02790325.  Google Scholar

[18]

C.-E. Pfister and W. Sullivan, On the topological entropy of saturated sets, Ergod. Th. Dynamical Sys., 27 (2007), 929-956.  doi: 10.1017/S0143385706000824.  Google Scholar

[19]

D. Ruelle, Statisticle mechanics on a compact set with $Z^{ν}$ actions satisfying expansiveness and specification, Trans. Amer. Math. Soc., 187 (1973), 237-251.   Google Scholar

[20]

K. Sigmund, Generic properties of invariant measures for Axiom A-diffeomorphisms, Invent. Math., 11 (1970), 99-109.  doi: 10.1007/BF01404606.  Google Scholar

[21]

K. Sigmund, On dynamical systems with specification property, Trans, Amer, Math. Soc., 190 (1974), 285-299.  doi: 10.1090/S0002-9947-1974-0352411-X.  Google Scholar

[22]

D. Tompson, Irregular sets, the beta-transformation and the almost specification property, Trans. Am. Math. Soc., 364 (2012), 5395-5414.  doi: 10.1090/S0002-9947-2012-05540-1.  Google Scholar

[23]

T. Ward and Q. Zhang, The Abramov-Rokhlin entropy addition formular for amenable group actions, Monatsh. Math., 114 (1992), 317-329.  doi: 10.1007/BF01299386.  Google Scholar

[24]

B. Weiss, Monotileable amenable groups, Topology, Ergodic Theory, Real Algebraic Geometry, in: Amer. Math. Soc. Transl. Ser. 2, vol. 202, Amer. Math. Soc., Providence, RI, (2001), 257-262.   Google Scholar

[25]

D. Zheng, E. Chen and J. Yang, On large deviations for amenable group actions, Discrete Contin. Dyn. Syst., 36 (2016), 7191-7206, arXiv:1507.05130. doi: 10.3934/dcds.2016113.  Google Scholar

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