Some Notes on Variational Principle for Metric Mean Dimension

Abstract

This paper investigates a variational principle for the Bowen metric mean dimension of saturated sets G K , where K is a compact connected subset of the convex combination of finite invariant measures for the systems with g-almost product property. In fact, we prove the variational principle of a saturated set with more information, that is G K ∩ {x ∈ X : C f (X) ⊂ ω f (x)}, which reveals that the limit point set of a saturated set contains all structure of the orbits. As an application, we obtain a more general version of multifractal analysis, which is derived independently and can imply partial results of

Bowen’s equations for upper metric mean dimension with potential

“…Construction of the Moran-like Fractal. The construction of the Moran-like Fractal is a standard process that inspired by Thompson [TH10] and Backes [BR23]. By [SHI22, Lemma 6], there exists a finite open cover U of X that satisfies diam(U) ≤ 5ǫ 0 , Leb(U) ≥ 5ǫ 0 4 .…”

A convex analysis approach to the metric mean dimension: Limits of scaled pressures and variational principles

“…For every γ > 0 and ε > 0, consider the probability measure µ γ ε (i) in (Y p ) N defined in (30), within the proof of Lemma 8.2, but now with respect to the fixed point Φ(σ i (ξ)) by σ. Then,…”

“…It is known that this order can be exchanged under the marker property (cf. [17,25,29] and also [30,31]), but there are relevant systems without this property, as shown in [26,21]. Regarding this issue, we refer the reader to Example 10.3 and Remark 10.4.…”