A Variational Principle for the Metric Mean Dimension of Level Sets
Lucas Backes, Fagner B. Rodrigues
Abstract
We prove a variational principle for the upper and lower metric mean dimension of level sets <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\left \{{x\in X: \lim _{n\to \infty }\frac {1}{n}\sum _{j=0}^{n-1}\varphi (f^{j}(x))=\alpha }\right \}$ </tex-math></inline-formula> associated to continuous potentials <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\varphi:X\to \mathbb R$ </tex-math></inline-formula> and continuous dynamics <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f:X\to X$ </tex-math></inline-formula> 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.