Maps in Dimension One with Infinite Entropy

We give examples of endomorphisms in dimension one with infinite topological entropy which are $\alpha$-H"older and $(1,p)$-Sobolev for all $0\le \alpha <1$ and $1\le p<\mathrm{\infty }$. This is constructed within a family of endomorphisms with infinite topological entropy and which traverse all $\alpha$-H"older 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.