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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 and Continuous Dynamical Systems, 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.

[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.

[3]

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

[4]

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

[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.

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[15]

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

[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.

[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.

[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.

[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. 

[20]

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

[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.

[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.

[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.

[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. 

[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.

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.

[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.

[3]

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

[4]

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

[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.

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[15]

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

[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.

[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.

[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.

[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. 

[20]

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

[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.

[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.

[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.

[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. 

[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.

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