Minimal spaces with cyclic group of homeomorphisms

There are two main subjects in this paper.

For a topological dynamical system $(X, T)$ we study the topological entropy of its “functional envelopes” (the action of $T$ by left composition on the space of all continuous self-maps or on the space of all self-homeomorphisms of $X$ ). In particular we prove that for zero-dimensional spaces $X$ both entropies are infinite except when $T$ is equicontinuous (then both equal zero).
We call $S l o v a k$ $s p a c e$ any compact metric space whose homeomorphism group is cyclic and generated by a minimal homeomorphism. Using Slovak spaces we provide examples of (minimal) systems $(X, T)$ with positive entropy, yet, whose functional envelope on homeomorphisms has entropy zero (answering a question posed by Kolyada and Semikina). Finally, also using Slovak spaces, we resolve a long standing open problem whether the circle is a unique non-degenerate continuum admitting minimal continuous transformations but only invertible: No, some Slovak spaces are such, as well.

There are two main subjects in this paper.

For a topological dynamical system $(X, T)$ we study the topological entropy of its “functional envelopes” (the action of $T$ by left composition on the space of all continuous self-maps or on the space of all self-homeomorphisms of $X$ ). In particular we prove that for zero-dimensional spaces $X$ both entropies are infinite except when $T$ is equicontinuous (then both equal zero).
We call $S l o v a k$ $s p a c e$ any compact metric space whose homeomorphism group is cyclic and generated by a minimal homeomorphism. Using Slovak spaces we provide examples of (minimal) systems $(X, T)$ with positive entropy, yet, whose functional envelope on homeomorphisms has entropy zero (answering a question posed by Kolyada and Semikina). Finally, also using Slovak spaces, we resolve a long standing open problem whether the circle is a unique non-degenerate continuum admitting minimal continuous transformations but only invertible: No, some Slovak spaces are such, as well.

文献摘要