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October  2021, 41(10): 4593-4608. doi: 10.3934/dcds.2021050

Variational relations for metric mean dimension and rate distortion dimension

LCSM (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, P. R. China

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

Fund Project: This work was supported by National Nature Science Foundation of China (12001192)

Recently, Lindenstrauss and Tsukamoto established a double variational principle between mean dimension theory and rate distortion theory. The main purpose of this paper is to develop some new variational relations for the metric mean dimension and the rate distortion dimension. Inspired by the dimension theory of topological entropy, we introduce and explore the Bowen metric mean dimension of subsets. Besides, we give some new characterizations for the rate distortion dimension. Finally, the relation between the Bowen metric mean dimension of the set of generic points and the rate distortion dimension is also investigated.

Citation: Tao Wang. Variational relations for metric mean dimension and rate distortion dimension. Discrete and Continuous Dynamical Systems, 2021, 41 (10) : 4593-4608. doi: 10.3934/dcds.2021050
References:
[1]

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

[2]

R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136.  doi: 10.1090/S0002-9947-1973-0338317-X.

[3]

M. Brin and A. Katok, On local entropy, in Geometric Dynamics, Rio de Janeiro, (1981), in Lecture Notes in Math., Springer, Berlin, 1007 (1983), 30–38. doi: 10.1007/BFb0061408.

[4]

E. Chen, D. Dou and D. Zheng, Variational principles for amenable metric mean dimensions, preprint, arXiv: 1708.02087, (2017).

[5]

D.-J. Feng and W. Huang, Variational principles for topological entropies of subsets, J. Funct. Anal., 263 (2012), 2228-2254.  doi: 10.1016/j.jfa.2012.07.010.

[6]

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.

[7]

A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, IHES Pub., 51 (1980), 137-173. 

[8]

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

[9]

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.

[10]

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

[11] P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge University Press, 1995.  doi: 10.1017/CBO9780511623813.
[12] Y. B. Pesin, Dimension theory in dynamical systems, Contemporary Views and Applications, University of Chicago Press, Chicago, 1997.  doi: 10.7208/chicago/9780226662237.001.0001.
[13]

C.-E. Pfister and W. G. Sullivan, On the topological entropy of saturated sets, Ergodic Theory Dynam. Systems, 27 (2007), 929-956.  doi: 10.1017/S0143385706000824.

[14]

M. Tsukamoto, Double variational principle for mean dimension with potential, Adv. Math., 361 (2020), 106935, 53 pp. doi: 10.1016/j.aim.2019.106935.

[15]

A. Velozo and R. Velozo, Rate distortion theory, metric mean dimension and measure theoretic entropy, arXiv: 1707.05762.

[16]

P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York–Berlin, 1982.

show all references

References:
[1]

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

[2]

R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136.  doi: 10.1090/S0002-9947-1973-0338317-X.

[3]

M. Brin and A. Katok, On local entropy, in Geometric Dynamics, Rio de Janeiro, (1981), in Lecture Notes in Math., Springer, Berlin, 1007 (1983), 30–38. doi: 10.1007/BFb0061408.

[4]

E. Chen, D. Dou and D. Zheng, Variational principles for amenable metric mean dimensions, preprint, arXiv: 1708.02087, (2017).

[5]

D.-J. Feng and W. Huang, Variational principles for topological entropies of subsets, J. Funct. Anal., 263 (2012), 2228-2254.  doi: 10.1016/j.jfa.2012.07.010.

[6]

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.

[7]

A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, IHES Pub., 51 (1980), 137-173. 

[8]

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

[9]

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.

[10]

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

[11] P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge University Press, 1995.  doi: 10.1017/CBO9780511623813.
[12] Y. B. Pesin, Dimension theory in dynamical systems, Contemporary Views and Applications, University of Chicago Press, Chicago, 1997.  doi: 10.7208/chicago/9780226662237.001.0001.
[13]

C.-E. Pfister and W. G. Sullivan, On the topological entropy of saturated sets, Ergodic Theory Dynam. Systems, 27 (2007), 929-956.  doi: 10.1017/S0143385706000824.

[14]

M. Tsukamoto, Double variational principle for mean dimension with potential, Adv. Math., 361 (2020), 106935, 53 pp. doi: 10.1016/j.aim.2019.106935.

[15]

A. Velozo and R. Velozo, Rate distortion theory, metric mean dimension and measure theoretic entropy, arXiv: 1707.05762.

[16]

P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York–Berlin, 1982.

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