6 problems

Let $(\mathbb{M},f)$ be a dynamical system, where $\mathbb{M}$ is a compact metric space and $f$ is its dynamics. Write $\operatorname{mdim}(\mathbb{M},f)$ for the mean topological…

Let $\mathcal{X}$ be a compact metrizable space and let $T\colon\mathbb{R}^d\times\mathcal{X}\to\mathcal{X}$ be a continuous action. Let $\varphi\colon\mathcal{X}\to\mathbb{R}$ be…

Metric characterization conjecture. The topological mean dimension is the infimum of the metric mean dimensions over all distances inducing the topology:

Metric realization conjecture. There always exists a compatible metric $d$ for which both equalities hold:

Metric mean dimension variational principle. Such a distance exists for every dynamical system $(\mathcal{X},T)$; that is, there is a distance $d$ on $\mathcal{X}$ satisfying

Metric realization conjecture. For every system $X$, there exists a distance $d$ such that