A variational principle for the metric mean dimension of level sets

Abstract

We prove a variational principle for the upper and lower metric mean dimension of level sets \[ \left\{x\in X: \lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}φ(f^{j}(x))=α\right\} \] associated to continuous potentials $\phi :X\to \mathbb{R}$ and continuous dynamics $f:X\to X$ defined on compact metric spaces and exhibiting the specification property. This result relates the upper and lower metric mean dimension of the above mentioned sets with growth rates of measure-theoretic entropy of partitions decreasing in diameter associated to some special measures. Moreover, we present several examples to which our result may be applied to. Similar results were previously known for the topological entropy and for the topological pressure.

To appear in IEEE Transactions on Information Theory

References in corpus (2)

• Around the variational principle for metric mean dimension
• On multifractal analysis and large deviations of singular-hyperbolic attractors

Cited by in corpus (2)

• The irregular set for maps with almost weak specification property has full metric mean dimension
• Variational principles for metric mean dimension with potential of level sets