Minimal Spaces with Cyclic Group of Homeomorphisms
| Autor: | Dariusz Tywoniuk, Tomasz Downarowicz, L'ubomír Snoha |
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| Rok vydání: | 2015 |
| Předmět: |
Discrete mathematics
37B05 (primary) ?37B40 ?54H20 (secondary) Pure mathematics Open problem 010102 general mathematics Cyclic group Dynamical Systems (math.DS) Topological entropy Homeomorphism group Equicontinuity 01 natural sciences 010305 fluids & plasmas law.invention Invertible matrix Compact space law 0103 physical sciences FOS: Mathematics Entropy (information theory) Mathematics - Dynamical Systems 0101 mathematics Analysis Mathematics |
| Zdroj: | Journal of Dynamics and Differential Equations. 29:243-257 |
| ISSN: | 1572-9222 1040-7294 |
| DOI: | 10.1007/s10884-015-9433-2 |
| Popis: | 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. 17 pages, 1 figure |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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