Upper metric mean dimensions with potential of $\epsilon$-stable sets

Abstract

It is well-known that $\epsilon$-stable sets have a deep connection with the topological entropy of dynamical systems. In the present paper, we investigate the relationships of three types of upper metric mean dimensions with potential between \emph{the blocks of $\epsilon$-stable sets, $\epsilon$-stable sets, the dispersion of preimages of $\epsilon$-stable sets} and the whole phase space. Besides, some chaotic phenomenons are revealed in infinite entropy systems. As an application of main results, we show tail entropy, preimage neighborhood entropy and topological entropy have the same metric mean dimension.