Flexibility of topological entropy and factor persistence in the $C^0$-Gromov Hausdorff topology

arXiv:2610.31939v1 Announce Type: new Abstract: We prove that every surjective topological dynamical system on a compact metric space can be approximated in the $C^0$-Gromov--Hausdorff topology by homeomorphisms with any prescribed topological entropy in $[0,+\infty]$, even when simultaneous approximation of its sets of invariant and ergodic measures is required. We also establish a rigidity result under uniform geometric bounds in a fixed dimension: for a homeomorphism of a closed, connected Riemannian manifold, topological stability in the sense of Gromov--Hausdorff guarantees its persistence as a factor of sufficiently close perturbations within the same geometric class. This persistence prevents topological entropy from decreasing and ensures that every invariant or ergodic measure of the original system admits a lift of the same type to the perturbed system.

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