Rate distortion theory, metric mean dimension and measure theoretic entropy

We prove a variational principle for the metric mean dimension analog to the one in [LT]. Instead of using the rate distortion function we use the function $h_{\mu }(ϵ,T,\delta )$ that is closely related to the entropy $h_{\mu }(T)$ of $\mu$. Our formulation has the advantage of being, in the authors opinion, more natural when doing computations. As a corollary we obtain a proof of the standard variational principle. We also obtain some relations between the rate distortion function with our function ${\tilde{h}}_{\mu }(ϵ,T,\delta )$, a modification of $h_{\mu }(ϵ,T,\delta )$ when replacing the dynamical metrics with the average dynamical metrics. Using our methods we also reprove the main result in [LT]. We will explain how to construct homeomorphisms on closed manifolds with maximal metric mean dimension. We end this paper with some questions that naturally arise from this work.