Maps in dimension one with infinite entropy

Abstract

For each real $\alpha , 0 \leq \alpha \lt 1$, we give examples of endomorphisms in dimension one with infinite topological entropy which are $\alpha$‑Hölder; and for each real $p , 1 \leq p\lt \infty$, we also give examples of endomorphisms in dimension one with infinite topological entropy which are $(1, p)$-Sobolev. These examples are constructed within a family of endomorphisms with infinite topological entropy and which traverse all $\alpha$-Hölder and $(1, p)$-Sobolev classes. Finally, we also give examples of endomorphisms, also in dimension one, which lie in the big and little Zygmund classes, answering a question of M. Benedicks.