Infinite entropy is generic in Hölder and Sobolev spaces (Q1681557)

A Lipschitz self-map on a compact metric space of finite Hausdorff dimension has finite topological entropy. On the other hand, in the space of homeomorphisms of a smooth compact \(d\)-manifold with \(d\geq2\) infinite topological entropy is generic in the \(C^0\)-topology by a result of \textit{K. Yano} [Invent. Math. 59, 215--220 (1980; Zbl 0434.54010)]. Here the question of what happens for degrees of smoothness in between the two results is addressed, specifically for the Hölder class \(C^{\alpha}\) with~\(\alpha\in[0,1)\) and the Sobolev class \(W^{1,p}\cap C^0\) with \(p\in[1,\infty)\) of continuous maps with weak derivatives in \(L^p\). The main results are of necessity somewhat technical, giving residual subsets with infinite topological entropy.