Bowen's equations for upper metric mean dimension with potential

Abstract

Firstly, we first introduce a new notion called induced upper metric mean dimension with potential, which naturally generalizes the definition of upper metric mean dimension with potential given by Tsukamoto to more general cases. Also, we establish a variational principle for it in terms of upper (and lower) rate distortion dimensions and show induced upper metric mean dimension with potential and upper metric mean dimension with potential are related by a Bowen equation. Secondly, we continue to introduce two new notions, called BS metric mean dimension and Packing BS metric mean dimension on arbitrary subsets, to establish Bowen's equations for upper metric mean dimension with potential on subsets. Besides, two corresponding variational principles for BS metric mean dimension and Packing BS metric mean dimension on subsets are obtained. Finally, the special interest about the upper metric mean dimension of generic points of ergodic measures are also involved.

Packing metric mean dimension of sets of generic points

“…Again, if µ ∈ E(X, T ), in the context of metric mean dimension Wang [W21] showed that the relationship between Bowen upper metric mean dimension of generic points of µ and its candidates used to describe it by an inequality. The authors [YCZ22] further verified that this inequality can be an equality under an increasing condition. In this paper, we obtain a new formula of packing metric mean dimension of generic points of ergodic measures without additional conditions compared with our previous work [YCZ22].…”

“…The authors [YCZ22] further verified that this inequality can be an equality under an increasing condition. In this paper, we obtain a new formula of packing metric mean dimension of generic points of ergodic measures without additional conditions compared with our previous work [YCZ22]. Additionally, we obtain the following Theorem 1.3 (Packing upper metric mean dimension of generic points of ergodic measures).…”

“…Then we establish two new variational principles for Bowen upper metric mean dimension and packing upper metric mean dimension on subsets in terms of two measure-theoretic quantities related to Katok's entropy defined by Carathēodory-Pesin structures. Finally, we obtain a new formula of packing metric mean dimension of generic points of ergodic measures without additional conditions compared with our previous work [YCZ22]. Additionally, we obtain inverse variational principles for packing metric mean dimension.…”

“…We would like to remark that the lower Katok's entropy can be considered a new candidate that we don not use in [YCZ22] to describe the metric mean dimension of generic points of ergodic measures.…”