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Scale pressure for amenable group actions
School of Mathematics, Northwest University, Xi'an, 710127, China |
In this paper we introduce the notion of scale pressure and measure theoretic scale pressure for amenable group actions. A variational principle for amenable group actions is presented. We also describe these quantities by pseudo-orbits. Moreover, we prove that if $ G $ is a finitely generated countable discrete amenable group, then the scale pressure of $ G $ coincides with the scale pressure of $ G $ with respect to pseudo-orbits.
References:
[1] |
N. P. Chung and K. Lee,
Topological stability and pseudo-orbit tracing property of group actions, P. Am. Math. Soc., 146 (2018), 1047-1057.
doi: 10.1090/proc/13654. |
[2] |
T. Downarowicz, B. Frej and P. P. Romagnoli, Shearer's inequality and Infimum Rule for Shannon entropy and topological entropy, in Contributions to Dynamics and numbers, American Mathematical Society, 2016.
doi: 10.1090/conm/669/13423. |
[3] |
M. Gromov,
Topological invariants of dynamical systems and spaces of holomorphic maps, I. Math. Phys. Anal. Geom., 2 (1999), 323-415.
doi: 10.1023/A:1009841100168. |
[4] |
W. Huang, X. Ye and G. Zhang,
Local entropy theory for a countable discrete amenable group action, J. Funct. Anal., 261 (2010), 1028-1082.
doi: 10.1016/j.jfa.2011.04.014. |
[5] |
E. Lindenstrauss and M. Tsukamoto,
From rate distortion theory to metric mean dimension: variational principle, IEEE T. Inform. Theory, 64 (2018), 3590-3609.
doi: 10.1109/TIT.2018.2806219. |
[6] |
E. Lindenstrauss,
Pointwise theorems for amenable groups, Invent. Math., 146 (2001), 259-295.
doi: 10.1007/s002220100162. |
[7] |
E. Lindenstrauss and M. Tsukamoto.,
Double variational principle for mean dimension, Geom. Funct. Anal., 29 (2019), 1048-1109.
doi: 10.1007/s00039-019-00501-8. |
[8] |
E. Lindenstrauss and B. Weiss,
Mean topological dimension, Israel J. Math., 115 (2000), 1-24.
doi: 10.1007/BF02810577. |
[9] |
T. Meyerovitch,
Pseudo-orbit tracing and algebraic actions of countable amenable groups, Ergod. Theor. Dyn. Syst., 39 (2019), 2570-2591.
doi: 10.1017/etds.2017.126. |
[10] |
D. S. Ornstein and B. Weiss,
Entropy and isomorphism theorems for actions of amenable groups, J. Anal. Math., 48 (1987), 1-141.
doi: 10.1007/BF02790325. |
[11] |
M. Tsukamoto, Double variational principle for mean dimension with potential, Adv. Math., 361 (2020), 106935.
doi: 10.1016/j.aim.2019.106935. |
[12] |
A. Velozo and R. Velozo, Rate distortion theory, metric mean dimension and measure theoretic entropy, preprint, arXiv: math/1707.05762. Google Scholar |
[13] |
B. Weiss, Actions of amenable groups, Topics in Dynamics and Ergodic Theory, Cambridge Univ. Press, 2003.
doi: 10.1017/CBO9780511546716.012.![]() ![]() |
[14] |
Y. Zhao,
Measure-theoretic pressure for amenable group actions, Colloq. Math., 148 (2017), 87-106.
doi: 10.4064/cm6784-6-2016. |
[15] |
D. Zheng, E. Chen and J. Yang,
On large deviations for amenable group actions, Discrete Contin. Dynam. Systems, 36 (2016), 7191-7206.
doi: 10.3934/dcds.2016113. |
[16] |
Y. 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. |
[17] |
K. Yano,
A remark on the topological entropy of homeomorphisms, Invent. Math., 59 (1980), 215-220.
doi: 10.1007/BF01453235. |
show all references
References:
[1] |
N. P. Chung and K. Lee,
Topological stability and pseudo-orbit tracing property of group actions, P. Am. Math. Soc., 146 (2018), 1047-1057.
doi: 10.1090/proc/13654. |
[2] |
T. Downarowicz, B. Frej and P. P. Romagnoli, Shearer's inequality and Infimum Rule for Shannon entropy and topological entropy, in Contributions to Dynamics and numbers, American Mathematical Society, 2016.
doi: 10.1090/conm/669/13423. |
[3] |
M. Gromov,
Topological invariants of dynamical systems and spaces of holomorphic maps, I. Math. Phys. Anal. Geom., 2 (1999), 323-415.
doi: 10.1023/A:1009841100168. |
[4] |
W. Huang, X. Ye and G. Zhang,
Local entropy theory for a countable discrete amenable group action, J. Funct. Anal., 261 (2010), 1028-1082.
doi: 10.1016/j.jfa.2011.04.014. |
[5] |
E. Lindenstrauss and M. Tsukamoto,
From rate distortion theory to metric mean dimension: variational principle, IEEE T. Inform. Theory, 64 (2018), 3590-3609.
doi: 10.1109/TIT.2018.2806219. |
[6] |
E. Lindenstrauss,
Pointwise theorems for amenable groups, Invent. Math., 146 (2001), 259-295.
doi: 10.1007/s002220100162. |
[7] |
E. Lindenstrauss and M. Tsukamoto.,
Double variational principle for mean dimension, Geom. Funct. Anal., 29 (2019), 1048-1109.
doi: 10.1007/s00039-019-00501-8. |
[8] |
E. Lindenstrauss and B. Weiss,
Mean topological dimension, Israel J. Math., 115 (2000), 1-24.
doi: 10.1007/BF02810577. |
[9] |
T. Meyerovitch,
Pseudo-orbit tracing and algebraic actions of countable amenable groups, Ergod. Theor. Dyn. Syst., 39 (2019), 2570-2591.
doi: 10.1017/etds.2017.126. |
[10] |
D. S. Ornstein and B. Weiss,
Entropy and isomorphism theorems for actions of amenable groups, J. Anal. Math., 48 (1987), 1-141.
doi: 10.1007/BF02790325. |
[11] |
M. Tsukamoto, Double variational principle for mean dimension with potential, Adv. Math., 361 (2020), 106935.
doi: 10.1016/j.aim.2019.106935. |
[12] |
A. Velozo and R. Velozo, Rate distortion theory, metric mean dimension and measure theoretic entropy, preprint, arXiv: math/1707.05762. Google Scholar |
[13] |
B. Weiss, Actions of amenable groups, Topics in Dynamics and Ergodic Theory, Cambridge Univ. Press, 2003.
doi: 10.1017/CBO9780511546716.012.![]() ![]() |
[14] |
Y. Zhao,
Measure-theoretic pressure for amenable group actions, Colloq. Math., 148 (2017), 87-106.
doi: 10.4064/cm6784-6-2016. |
[15] |
D. Zheng, E. Chen and J. Yang,
On large deviations for amenable group actions, Discrete Contin. Dynam. Systems, 36 (2016), 7191-7206.
doi: 10.3934/dcds.2016113. |
[16] |
Y. 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. |
[17] |
K. Yano,
A remark on the topological entropy of homeomorphisms, Invent. Math., 59 (1980), 215-220.
doi: 10.1007/BF01453235. |
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