A variational principle for metric mean dimension via lower Brin-Katok local entropy

We prove a finite-scale comparison between lower Brin-Katok local entropy and Katok covering entropy. Let $(\mathcal{X},d,T)$ be a compact metric topological dynamical system and let $\mu$ be ergodic. Then, for every $\epsilon >0$ and every $\delta \in (0,1)$, $$h_{\mu }^K(6\epsilon ,\delta )\le {\underset{‾}{h}}_{\mu }^{BK}(\epsilon )\mathrm{.}$$ Combining this estimate with the usual Katok-type variational principle for metric mean dimension gives the corresponding variational principle with lower Brin-Katok local entropy.