Minimal Spaces with Cyclic Group of Homeomorphisms

• Published in: Journal of dynamics and differential equations Vol. 29; no. 1; pp. 243 - 257
• Main Authors: Downarowicz, Tomasz, Snoha, L’ubomír, Tywoniuk, Dariusz
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.03.2017; Springer Nature B.V
• Subjects: Applications of Mathematics; Continuity (mathematics); Entropy; Mathematics; Mathematics and Statistics; Metric space; Ordinary Differential Equations; Partial Differential Equations; Continuum; 37B40; Minimal space; 54H20 (Secondary); Topological entropy; Group of homeomorphisms; Functional envelope; 37B05 (Primary); Almost 1-1 extension
• ISSN: 1040-7294; 1572-9222
• DOI: 10.1007/s10884-015-9433-2

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.