Scale pressure for amenable group actions

Dandan Cheng; Qian Hao; Zhiming Li

doi:10.3934/cpaa.2021008

Article Contents

2021, Volume 20, Issue 3: 1091-1102. Doi: 10.3934/cpaa.2021008

This issue Previous Article Rational limit cycles of Abel equations Next Article Boundary-Domain Integral Equations equivalent to an exterior mixed BVP for the variable-viscosity compressible Stokes PDEs

Scale pressure for amenable group actions

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

^* Corresponding author
^* Corresponding author

Received: June 30, 2020

Revised: October 31, 2020

Early access: January 2021

Published: March 2021

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

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 37A35; Secondary: 37B40.

Citation:

Full Text(HTML)

Related Papers

Cited by

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.