American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Low Mach number limit for the compressible magnetohydrodynamic equations in a periodic domain
  • DCDS Home
  • This Issue
  • Next Article
    Weakly mixing diffeomorphisms preserving a measurable Riemannian metric with prescribed Liouville rotation behavior
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

[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

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

[1]

Xiaojun Huang, Yuan Lian, Changrong Zhu. A Billingsley-type theorem for the pressure of an action of an amenable group. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 959-993. doi: 10.3934/dcds.2019040
[2]

Kazumine Moriyasu, Kazuhiro Sakai, Kenichiro Yamamoto. Regular maps with the specification property. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2991-3009. doi: 10.3934/dcds.2013.33.2991
[3]

Welington Cordeiro, Manfred Denker, Xuan Zhang. On specification and measure expansiveness. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1941-1957. doi: 10.3934/dcds.2017082
[4]

Welington Cordeiro, Manfred Denker, Xuan Zhang. Corrigendum to: On specification and measure expansiveness. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3705-3706. doi: 10.3934/dcds.2018160
[5]

Yoshikazu Katayama, Colin E. Sutherland and Masamichi Takesaki. The intrinsic invariant of an approximately finite dimensional factor and the cocycle conjugacy of discrete amenable group actions. Electronic Research Announcements, 1995, 1: 43-47.

[6]

Dongmei Zheng, Ercai Chen, Jiahong Yang. On large deviations for amenable group actions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7191-7206. doi: 10.3934/dcds.2016113
[7]

Brandon Seward. Every action of a nonamenable group is the factor of a small action. Journal of Modern Dynamics, 2014, 8 (2) : 251-270. doi: 10.3934/jmd.2014.8.251
[8]

Jinjun Li, Min Wu. Divergence points in systems satisfying the specification property. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 905-920. doi: 10.3934/dcds.2013.33.905
[9]

Lin Wang, Yujun Zhu. Center specification property and entropy for partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 469-479. doi: 10.3934/dcds.2016.36.469
[10]

Xiaojun Huang, Jinsong Liu, Changrong Zhu. The Katok's entropy formula for amenable group actions. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4467-4482. doi: 10.3934/dcds.2018195
[11]

Jinjun Li, Min Wu. Generic property of irregular sets in systems satisfying the specification property. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 635-645. doi: 10.3934/dcds.2014.34.635
[12]

S. A. Krat. On pairs of metrics invariant under a cocompact action of a group. Electronic Research Announcements, 2001, 7: 79-86.

[13]

Manfred G. Madritsch, Izabela Petrykiewicz. Non-normal numbers in dynamical systems fulfilling the specification property. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4751-4764. doi: 10.3934/dcds.2014.34.4751
[14]

Lidong Wang, Hui Wang, Guifeng Huang. Minimal sets and $\omega$-chaos in expansive systems with weak specification property. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1231-1238. doi: 10.3934/dcds.2015.35.1231
[15]

Jean-Paul Thouvenot. The work of Lewis Bowen on the entropy theory of non-amenable group actions. Journal of Modern Dynamics, 2019, 15: 133-141. doi: 10.3934/jmd.2019016
[16]

Carlos Matheus, Jean-Christophe Yoccoz. The action of the affine diffeomorphisms on the relative homology group of certain exceptionally symmetric origamis. Journal of Modern Dynamics, 2010, 4 (3) : 453-486. doi: 10.3934/jmd.2010.4.453
[17]

Alessandro Fonda, Antonio J. Ureña. Periodic, subharmonic, and quasi-periodic oscillations under the action of a central force. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 169-192. doi: 10.3934/dcds.2011.29.169
[18]

Eldho K. Thomas, Nadya Markin, Frédérique Oggier. On Abelian group representability of finite groups. Advances in Mathematics of Communications, 2014, 8 (2) : 139-152. doi: 10.3934/amc.2014.8.139
[19]

Rostyslav Kravchenko. The action of finite-state tree automorphisms on Bernoulli measures. Journal of Modern Dynamics, 2010, 4 (3) : 443-451. doi: 10.3934/jmd.2010.4.443
[20]

Evgeny L. Korotyaev. Estimates for solutions of KDV on the phase space of periodic distributions in terms of action variables. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 219-225. doi: 10.3934/dcds.2011.30.219

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (133)
  • HTML views (384)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences