Behavior of measures of maximal entropy under C^0–Gromov–Hausdorff perturbations

Investigate the behavior of measures of maximal entropy for homeomorphisms of compact metric spaces under perturbations in the $C^0$–Gromov–Hausdorff topology, determining which properties (e.g., existence, uniqueness, and support) are preserved or how they vary under such perturbations.

Background

The paper demonstrates that topological entropy is not stable under $C^0$–Gromov–Hausdorff perturbations by showing approximation by zero-entropy systems within a broad class of dynamics. This raises natural questions about finer measure-theoretic invariants.

Measures of maximal entropy (MMEs) are central in thermodynamic formalism and entropy theory. Understanding how MMEs behave when systems are perturbed in the $C^0$–Gromov–Hausdorff sense—where even the phase spaces may vary—is an unresolved issue highlighted by the authors.