C0-robustness of topological entropy for geodesic flows - Alves, Marcelo R. R.; Dahinden, Lucas; Meiwes, Matthias; Merlin, Louis (Journal of fixed point theory and applications, 2022)
In this paper, we study the regularity of topological entropy, as a function on the space of Riemannian metrics endowed with the C0 topology. We establish several instances of entropy robustness (persistence of positive entropy after small C0 perturbations). A large part of this paper is dedicated to metrics on the two-dimensional torus, for which our main results are that metrics with a contractible closed geodesic have robust entropy (thus, generalizing and quantifying a result of Denvir-Mackay) and that metrics with robust positive entropy on the torus are C infty generic. Moreover, we quantify the asymptotic behavior of volume entropy in the Teichmüller space of hyperbolic metrics on a punctured torus, which bounds from below the topological entropy for these metrics. For general closed manifolds of dimension at least 2, we prove that the set of metrics with robust and large positive entropy is C0-large in the sense that it is dense and contains cones and arbitrarily large balls.
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Gesehen am 20.05.2022