  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.
