Minimal Spaces with Cyclic Group of Homeomorphisms - Downarowicz, Tomasz; Snoha, L’ubomír; Tywoniuk, Dariusz (Journal of dynamics and differential equations, 2017)
There are two main subjects in this paper. (1) 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). (2) We call Slovak space 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.