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March  2021, 20(3): 1091-1102. doi: 10.3934/cpaa.2021008

Scale pressure for amenable group actions

Dandan Cheng , Qian Hao and  Zhiming Li ,

School of Mathematics, Northwest University, Xi'an, 710127, China

* Corresponding author

Received  July 2020 Revised  November 2020 Published  March 2021 Early access  January 2021

Fund Project: The third author is supported by NSFC (No.11871394), and Natural Science Foundation of Shaanxi Province (2020JC-39)

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.

Keywords: Scale pressure, Variational principle, amenable group, Pseudo-orbits.
Mathematics Subject Classification: Primary: 37A35; Secondary: 37B40.
Citation: Dandan Cheng, Qian Hao, Zhiming Li. Scale pressure for amenable group actions. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1091-1102. doi: 10.3934/cpaa.2021008
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.

[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.

[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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