Genericity of Infinite Entropy for Maps with Low Regularity

TL;DR

This paper demonstrates that in certain low-regularity function spaces, the typical behavior of maps on compact manifolds is to have infinite topological entropy, contrasting with the finite entropy of bi-Lipschitz maps.

Contribution

It establishes that infinite entropy is generic in the closure of bi-Lipschitz homeomorphisms under H"older and Sobolev topologies, and extends the $C^1$-Closing Lemma to these spaces.

Findings

Topological entropy is generically infinite in the H"older and Sobolev closures of bi-Lipschitz homeomorphisms.

Examples of homeomorphisms with infinite entropy exist in all H"older and Sobolev classes.

Versions of the $C^1$-Closing Lemma are proved for these low-regularity spaces.

Abstract

For bi-Lipschitz homeomorphisms of a compact manifold it is known that topological entropy is always finite. For compact manifolds of dimension two or greater, we show that in the closure of the space of bi-Lipschitz homeomorphisms, with respect to either the H\"older or the Sobolev topologies, topological entropy is generically infinite. We also prove versions of the $C^1$-Closing Lemma in either of these spaces. Finally, we give examples of homeomorphisms with infinite topological entropy which are H\"older and/or Sobolev of every exponent.