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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 & Continuous Dynamical Systems, 2021, 41 (10) : 4593-4608. doi: 10.3934/dcds.2021050
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