2015/03/11 by Tomasz Downarowicz, Downarowicz, Tomasz, Ľubomír Snoha +3
Mathematics · #37B05 (primary) ?37B40 #?54H20 (secondary) #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.1503.03246
openalex publication_date 2015/03/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.