Minimal spaces with cyclic group of homeomorphisms

Downarowicz, Tomasz Minimal spaces with cyclic group of homeomorphisms / Tomasz Downarowicz, Ľubomír Snoha, Dariusz Tywoniuk. -- There are two main subjects in this paper. (1) For a topological dynamical system we study the topological entropy of its "functional envelopes" (the action of by left composition on the space of all continuous self-maps or on the space of all self-homeomorphisms of ). In particular we prove that for zero-dimensional spaces both entropies are infinite except when 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 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.

In Journal of Dynamics and Differential Equations. -- New York : Springer, 2017. -- ISSN 1040-7294. -- ISSN 1572-9222. -- Vol. 29, no. 1 (2017), pp. 243-257

1. Continuum (Mathematics) 2. homeomorphisms 3. topologická entropia

I. Snoha, Ľubomír, 1955-
II. Tywoniuk, Dariusz
III. Journal of Dynamics and Differential Equations. -- Vol. 29, no. 1 (2017), pp. 243-257

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