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Double variational principle conjecture for mean dimension with potential

Establish the double variational principle for mean dimension with potential: for any continuous action T: R^k × X → X on a compact metrizable space X and any continuous potential function φ: X → R, construct a metric d on X compatible with the topology such that mdim(X, T, φ) = sup_{μ ∈ M^T(X)} [(X, T, d, μ) + ∫_X φ dμ] = mdim(X, T, d, φ).

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Background

The paper proves a variational principle for mean dimension with potential that yields inequalities relating topological and metric quantities. In Remark (double variational principle conjecture), the author points out that, unlike the classical pressure variational principle which gives equalities, the current framework yields only inequalities. The conjecture proposes that one can select a compatible metric so that the inequalities become equalities universally for all compact metrizable spaces and continuous Rk-actions, thus extending the classical spirit of variational principles to mean dimension with potential.

The author later shows that this conjecture holds for the specific dynamical system of Brody curves with certain natural potentials, but the general case remains unresolved and is stated explicitly as a conjecture.

References

Indeed, we conjecture that for any continuous action T\colon \mathbb{R}k\times \mathcal{X}\to \mathcal{X} on a compact metrizable space \mathcal{X} and for any continuous function \varphi\colon \mathcal{X}\to \mathbb{R} there exists a metric \mathbf{d} on \mathcal{X} compatible with the given topology and satisfying \begin{equation*} \begin{split} mdim\left(\mathcal{X}, T, \varphi\right) &= \sup_{\mu \in \mathscr{M}T(\mathcal{X})} \left(\left(\mathcal{X}, T, \mathbf{d}, \mu\right) + \int_{\mathcal{X}\varphi\, d\mu\right) \ & = \sup_{\mu \in \mathscr{M}T(\mathcal{X})} \left(\left(\mathcal{X}, T, \mathbf{d}, \mu\right) + \int_{\mathcal{X}\varphi\, d\mu\right) = mdim\left(\mathcal{X}, T, \mathbf{d}, \varphi\right). \end{split} \end{equation*} This is more satisfactory (if it is true).

— Rate distortion dimension of random Brody curves (2403.11442 - Tsukamoto, 18 Mar 2024) in Remark (double variational principle conjecture), Section 4.1 (Definitions and variational principle)