Double Variational Principle of Mean Dimension for ${\mathbb {Z}}^{k}$-Actions

Abstract

Lindenstrauss and Tsukamoto in 2019 established double variational principle for mean dimension. In this paper, we focus on developing the mean dimension theory for ${\mathbb {Z}}^k$-actions. Specifically, we establish a double variational principle for mean dimension of ${\mathbb {Z}}^k$-actions for dynamical systems with the marker property.

Keywords

${\mathbb {Z}}^{k}$-action')">${\mathbb {Z}}^{k}$-action / Mean dimension / Rate distortion dimension / Double variational principle / 37A05 / 37B99

References

[1] AdlerR, KonheimA, McAndrewJ. Topological entropy. Trans. Amer. Math. Soc., 1965, 114: 309-319

[2] ChenE, DouD, ZhengD. Variational principle for amenable metric mean dimensions. J. Differential Equations, 2022, 319: 41-79

[3] CoornaertMTopological dimension and dynamical systems, 2015, New York. Springer.

[4] CoverT, ThomasJElements of information theory, 2006, New York. Wiley.

[5] DinaburgE. A correlation between topological entropy and metric entropy. Dokl. Akad. Nauk SSSR, 1970, 190: 19-22

[6] DownarowiczT. Minimal models for noninvertible and not uniquely ergodic systems. Israel J. Math., 2006, 156: 93-110

[7] GatzourasD, PeresY. Invariant measures of full dimension for some expanding maps. Ergodic Theory Dynam. Systems, 1997, 17: 147-167

[8] GoodmanTNT. Relating topological entropy and measure entropy. Bull. London Math. Soc., 1971, 3: 176-180

[9] GoodwynLW. Topological entropy bounds measure-theoretic entropy. Proc. Amer. Math. Soc., 1969, 23: 679-688

[10] GromovM. Topological invariants of dynamical systems and spaces of holomorphic maps, I. Math. Phys. Anal. Geom., 1999, 2: 323-415

[11] GutmanY. Embedding ${\mathbb{Z} }^{k}$-actions in cubical shifts and ${\mathbb{Z} }^{k}$-symbolic extensions. Ergodic Theory Dynam. Systems, 2011, 31: 383-403

[12] GutmanY, LindenstraussE, TsukamotoM. Mean dimension of ${\mathbb{Z} }^{k}$-actions. Geom. Funct. Anal., 2016, 26: 778-817

[13] GutmanY, QiaoY, TsukamotoM. Application of signal analysis to the embedding problem of ${\mathbb{Z} }^{k}$-actions. Geom. Funct. Anal., 2019, 29: 1440-1502

[14] GutmanY, TsukamotoM. Embedding minimal dynamical systems into Hilbert cubes. Invent. Math., 2020, 221: 113-166

[15] KawabataT, DemboA. The rate distortion dimension of sets and measures. IEEE Trans. Inform. Theory., 1994, 40: 1564-1572

[16] LiH. Sofic mean dimension. Adv. Math., 2013, 244: 570-604

[17] LiH, LiangB. Mean dimension, mean rank and von Neumann-Luck rank. J. Reine Angew. Math., 2018, 739: 207-240

[18] LindenstraussE. Mean dimension, small entropy factors and an embedding theorem. Publ. Math. Inst. Hautes Études Sci., 1999, 89: 227-262

[19] LindenstraussE, TsukamotoM. Mean dimension and an embedding problem: an example. Israel J. Math., 2014, 199: 573-584

[20] LindenstraussE, TsukamotoM. From rate distortion theory to metric mean dimension: variational principle. IEEE Trans. Inform. Theory., 2018, 64: 3590-3609

[21] LindenstraussE, TsukamotoM. Double variational principle for mean dimension. Geom. Funct. Anal., 2019, 29: 1048-1109

[22] LindenstraussE, WeissB. Mean topological dimension. Israel J. Math., 2000, 115: 1-24

[23] MisiurewiczM. A short proof of the variational principle for a ${\mathbb{Z} }_{+}^{N}$ action on a compact metric space. Asterisque, 1976, 40: 147-157

[24] MeyerovitchT, TsukamotoM. Expansive multiparameter actions and mean dimension. Trans. Amer. Math. Soc., 2019, 371: 7275-7299

[25] OrnseinD, WeissB. Entropy and isomorphism theorems for actions of amenable groups. J. Anal. Math., 1987, 48: 1-141

[26] RuelleD. Statistical mechanics on a compact set with ${\mathbb{Z} }^{v}$ action satisfying expansiveness and specification. Trans. Amer. Math. Soc., 1973, 185: 237-251

[27] Tsukamoto, M.: Double variational principle for mean dimension with potential. Adv. Math. 361 (2020)

[28] WangY, ChenE. Double variational principle for mean dimensions with sub-additive potentials. J. Math. Phys., 2023, 64082701 28

Funding

National Natural Science Foundation of China(12201328)

RIGHTS & PERMISSIONS

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature