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Register a new userOn relative metric mean dimension with potential and variational principles
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Weisheng Wu
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In this article, we introduce a notion of relative mean metric dimension with potential for a factor map $pi: (X,d, T)to (Y, S)$ between two topological dynamical systems. To link it with ergodic theory, we establish four variational principles in terms of metric entropy of partitions, Shapiras entropy, Katoks entropy and Brin-Katok local entropy respectively. Some results on local entropy with respect to a fixed open cover are obtained in the relative case. We also answer an open question raised by Shi cite{Shi} partially for a very well-partitionable compact metric space, and in general we obtain a variational inequality involving box dimension of the space. Corresponding inner variational principles given an invariant measure of $(Y,S)$ are also investigated.
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In this note, we show several variational principles for metric mean dimension. First we prove a variational principles in terms of Shapiras entropy related to finite open covers. Second we establish a variational principle in terms of Katoks entropy. Finally using these two variational principles we develop a variational principle in terms of Brin-Katok local entropy.
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