Rate distortion theory, metric mean dimension and measure theoretic entropy

By A. Velozo, Renato Velozo. 2017

64 citations recorded by the source corpus.

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 $h_\mu(\epsilon,T,\delta)$ 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 $\tilde{h}_\mu(\epsilon,T,\delta)$, a modification of $h_\mu(\epsilon,T,\delta)$ 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.

Subfields: Analytical Chemistry, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/14e0ea9e5a82c3f2cb05cf3ba8cdaa0e520d23bd

Citations in the corpus

• Bounded complexity, mean equicontinuity and discrete spectrum (2018, unknown): “uity,discrete spectrum. 1 2 W. HUANG, J. LI, J. THOUVENOT, L. XU AND X. YE Katok [18], by using the mean metric instead of the Bowen metric (for discussionand results related to mean metric, see also [23, 30]). They showed that if an invariant measure has discrete spectrum,then the measure complexit”