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Weakly mixing diffeomorphisms preserving a measurable Riemannian metric with prescribed Liouville rotation behavior
Periodic measures are dense in invariant measures for residually finite amenable group actions with specification
1. | School of Mathematical Sciences, Peking University, Beijing 100871, China |
2. | Department of Mathematics, SUNY at Buffalo, Buffalo, NY 14260-2900, USA |
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.
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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