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.

中文翻译：

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具有无限熵的第一维映射

对于每个真实的$ \ alpha，0 \ leq \ alpha \ lt 1 $，我们给出具有无限拓扑熵的第一维内同态的示例，即$ \ alpha $ ‑Hölder；对于每个实数$ p，1 \ leq p \ lt \ infty $，我们还给出了具有无限拓扑熵的一维内同态的示例，即$（1，p）$-Sobolev。这些示例在具有无限拓扑熵的内同态族内构造，并且遍历所有$ \ alpha $-Hölder和$（1，p）$-Sobolev类。最后，我们还给出了内维同构的例子，同样在第一个维度上，它存在于大小不一的Zygmund类中，回答了Benedicks先生的问题。