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doi: 10.3934/dcds.2022050
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Mean dimension theory in symbolic dynamics for finitely generated amenable groups

1. 

School of Science, Ningbo University of Technology, Ningbo 315211, Zhejiang China

2. 

School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210046, Jiangsu China

*Corresponding author: Xiaoyao Zhou

Received  September 2021 Revised  March 2022 Early access April 2022

In this paper, we mainly show a close relationship between topological entropy and mean dimension theory for actions of polynomial growth groups. We show that metric mean dimension and mean Hausdorff dimension of subshifts with respect to the lower rank subgroup are equal to its topological entropy multiplied by the growth rate of the subgroup. Meanwhile, we prove that above result holds for rate distortion dimension of subshifts with respect to a lower rank subgroup and measure entropy. Furthermore, we present some examples.

Citation: Yunping Wang, Ercai Chen, Xiaoyao Zhou. Mean dimension theory in symbolic dynamics for finitely generated amenable groups. Discrete and Continuous Dynamical Systems, doi: 10.3934/dcds.2022050
References:
[1]

R. AdlerA. Konheim and M. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319.  doi: 10.1090/S0002-9947-1965-0175106-9.

[2]

H. Bass, The degree of polynomial growth of finitely generated nilpotent groups, Proc. London Math. Soc., 25 (1972), 603-614.  doi: 10.1112/plms/s3-25.4.603.

[3]

T. Ceccherini-Silberstein and M. Coornaert, Cellular Automata and Groups, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-14034-1.

[4]

E. ChenD. Dou and D. Zheng, Variational principle for amenable metric mean dimensions, J. Differential Equations, 319 (2022), 41-79.  doi: 10.1016/j.jde.2022.02.046.

[5]

E. Chen and J. Xiong, Dimension and measure theoretic entropy of a subshift in symbolic space, Chinese Sci. Bull., 42 (1997), 1193-1196.  doi: 10.1007/BF02882845.

[6]

M. Coornaert, Topological Dimension and Dynamical Systems, Springer, Cham, 2015. doi: 10.1007/978-3-319-19794-4.

[7]

T. Cover and J. Thomas, Elements of Information Theory, 2$^{nd}$ edition, Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, 2006.

[8]

D. Dou and R. Zhang, A note on dimensional entropy for amenable group actions, Topol. Methods Nonlinear Anal., 51 (2018), 599-608.  doi: 10.12775/tmna.2017.056.

[9]

H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory., 1 (1967), 1-49.  doi: 10.1007/BF01692494.

[10]

D. Gatzouras and Y. Peres, Invariant measures of full dimension for some expanding maps, Ergodic Theory Dynam. Systems, 17 (1997), 147-167.  doi: 10.1017/S0143385797060987.

[11]

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.

[12]

M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math., 53 (1981), 53-73. 

[13]

Y. GutmanE. Lindenstrauss and M. Tsukamoto, Mean dimension of $\mathbb{Z}^{k}$-actions, Geom. Funct. Anal., 26 (2016), 778-817.  doi: 10.1007/s00039-016-0372-9.

[14]

Y. GutmanY. Qiao and M. Tsukamoto, Application of signal analysis to the embedding problem of $\mathbb{Z}^{k}$-actions, Geom. Funct. Anal., 29 (2019), 1440-1502.  doi: 10.1007/s00039-019-00499-z.

[15]

Y. Gutman and M. Tsukamoto, Embedding minimal dynamical systems into Hilbert cubes, Invent. Math., 221 (2020), 113-166.  doi: 10.1007/s00222-019-00942-w.

[16]

L. Jin and Y. Qiao, The Hilbert cube contains a minimal subshift of full mean dimension, preprint, 2021, arXiv: 2102.10339.

[17]

T. Kawabata and A. Dembo, The rate distortion dimension of sets and measures, IEEE Trans. Inform. Theory., 40 (1994), 1564-1572.  doi: 10.1109/18.333868.

[18]

D. Kerr and H. Li, Ergodic Theory: Independence and Dichotomies, Springer Monographs in Mathematics. Springer, Cham, 2016. doi: 10.1007/978-3-319-49847-8.

[19]

H. Li and B. Liang, Mean dimension, mean rank and von Neumann-Luck rank, J. Reine Angew. Math., 739 (2018), 207-240.  doi: 10.1515/crelle-2015-0046.

[20]

E. Lindenstrauss, Pointwise theorems for amenable groups, Invent. Math., 146 (2001), 259-295.  doi: 10.1007/s002220100162.

[21]

E. Lindenstrauss and M. Tsukamoto, Mean dimension and an embedding problem: An example, Israel J. Math., 199 (2014), 573-584.  doi: 10.1007/s11856-013-0040-9.

[22]

E. Lindenstrauss and M. Tsukamoto, From rate distortion theory to metric mean dimension: Variational principle, IEEE Trans. Inform. Theory., 64 (2018), 3590-3609.  doi: 10.1109/TIT.2018.2806219.

[23]

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.

[24]

E. Lindenstrauss and B. Weiss, Mean topological dimension, Israel J. Math., 115 (2000), 1-24.  doi: 10.1007/BF02810577.

[25]

T. Meyerovitch and M. Tsukamoto, Expansive multiparameter actions and mean dimension, Trans. Amer. Math. Soc., 371 (2019), 7275-7299.  doi: 10.1090/tran/7588.

[26]

D. Ornstein and B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Analyse Math., 48 (1987), 1-141.  doi: 10.1007/BF02790325.

[27]

M. Shinoda and M. Tsukamoto, Symbolic dynamics in mean dimension theory, Ergodic Theory Dynam. Systems, 41 (2021), 2542-2560.  doi: 10.1017/etds.2020.47.

[28]

S. Simpson, Symbolic dynamics: Entropy=dimension=complexity, Theory Comput. Syst., 56 (2015), 527-543.  doi: 10.1007/s00224-014-9546-8.

[29]

M. Tsukamoto, Mean dimension of the dynamical system of Brody curves, Invent. Math., 211 (2018), 935-968.  doi: 10.1007/s00222-017-0758-9.

show all references

References:
[1]

R. AdlerA. Konheim and M. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319.  doi: 10.1090/S0002-9947-1965-0175106-9.

[2]

H. Bass, The degree of polynomial growth of finitely generated nilpotent groups, Proc. London Math. Soc., 25 (1972), 603-614.  doi: 10.1112/plms/s3-25.4.603.

[3]

T. Ceccherini-Silberstein and M. Coornaert, Cellular Automata and Groups, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-14034-1.

[4]

E. ChenD. Dou and D. Zheng, Variational principle for amenable metric mean dimensions, J. Differential Equations, 319 (2022), 41-79.  doi: 10.1016/j.jde.2022.02.046.

[5]

E. Chen and J. Xiong, Dimension and measure theoretic entropy of a subshift in symbolic space, Chinese Sci. Bull., 42 (1997), 1193-1196.  doi: 10.1007/BF02882845.

[6]

M. Coornaert, Topological Dimension and Dynamical Systems, Springer, Cham, 2015. doi: 10.1007/978-3-319-19794-4.

[7]

T. Cover and J. Thomas, Elements of Information Theory, 2$^{nd}$ edition, Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, 2006.

[8]

D. Dou and R. Zhang, A note on dimensional entropy for amenable group actions, Topol. Methods Nonlinear Anal., 51 (2018), 599-608.  doi: 10.12775/tmna.2017.056.

[9]

H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory., 1 (1967), 1-49.  doi: 10.1007/BF01692494.

[10]

D. Gatzouras and Y. Peres, Invariant measures of full dimension for some expanding maps, Ergodic Theory Dynam. Systems, 17 (1997), 147-167.  doi: 10.1017/S0143385797060987.

[11]

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.

[12]

M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math., 53 (1981), 53-73. 

[13]

Y. GutmanE. Lindenstrauss and M. Tsukamoto, Mean dimension of $\mathbb{Z}^{k}$-actions, Geom. Funct. Anal., 26 (2016), 778-817.  doi: 10.1007/s00039-016-0372-9.

[14]

Y. GutmanY. Qiao and M. Tsukamoto, Application of signal analysis to the embedding problem of $\mathbb{Z}^{k}$-actions, Geom. Funct. Anal., 29 (2019), 1440-1502.  doi: 10.1007/s00039-019-00499-z.

[15]

Y. Gutman and M. Tsukamoto, Embedding minimal dynamical systems into Hilbert cubes, Invent. Math., 221 (2020), 113-166.  doi: 10.1007/s00222-019-00942-w.

[16]

L. Jin and Y. Qiao, The Hilbert cube contains a minimal subshift of full mean dimension, preprint, 2021, arXiv: 2102.10339.

[17]

T. Kawabata and A. Dembo, The rate distortion dimension of sets and measures, IEEE Trans. Inform. Theory., 40 (1994), 1564-1572.  doi: 10.1109/18.333868.

[18]

D. Kerr and H. Li, Ergodic Theory: Independence and Dichotomies, Springer Monographs in Mathematics. Springer, Cham, 2016. doi: 10.1007/978-3-319-49847-8.

[19]

H. Li and B. Liang, Mean dimension, mean rank and von Neumann-Luck rank, J. Reine Angew. Math., 739 (2018), 207-240.  doi: 10.1515/crelle-2015-0046.

[20]

E. Lindenstrauss, Pointwise theorems for amenable groups, Invent. Math., 146 (2001), 259-295.  doi: 10.1007/s002220100162.

[21]

E. Lindenstrauss and M. Tsukamoto, Mean dimension and an embedding problem: An example, Israel J. Math., 199 (2014), 573-584.  doi: 10.1007/s11856-013-0040-9.

[22]

E. Lindenstrauss and M. Tsukamoto, From rate distortion theory to metric mean dimension: Variational principle, IEEE Trans. Inform. Theory., 64 (2018), 3590-3609.  doi: 10.1109/TIT.2018.2806219.

[23]

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.

[24]

E. Lindenstrauss and B. Weiss, Mean topological dimension, Israel J. Math., 115 (2000), 1-24.  doi: 10.1007/BF02810577.

[25]

T. Meyerovitch and M. Tsukamoto, Expansive multiparameter actions and mean dimension, Trans. Amer. Math. Soc., 371 (2019), 7275-7299.  doi: 10.1090/tran/7588.

[26]

D. Ornstein and B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Analyse Math., 48 (1987), 1-141.  doi: 10.1007/BF02790325.

[27]

M. Shinoda and M. Tsukamoto, Symbolic dynamics in mean dimension theory, Ergodic Theory Dynam. Systems, 41 (2021), 2542-2560.  doi: 10.1017/etds.2020.47.

[28]

S. Simpson, Symbolic dynamics: Entropy=dimension=complexity, Theory Comput. Syst., 56 (2015), 527-543.  doi: 10.1007/s00224-014-9546-8.

[29]

M. Tsukamoto, Mean dimension of the dynamical system of Brody curves, Invent. Math., 211 (2018), 935-968.  doi: 10.1007/s00222-017-0758-9.

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