Topology and Topological Sequence Entropy
| Autor: | Ruifeng Zhang, L'ubomír Snoha, Xiangdong Ye |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2018 |
| Předmět: |
Sequence
General Mathematics 010102 general mathematics Dynamical Systems (math.DS) Topology 01 natural sciences Infimum and supremum 010101 applied mathematics Sequence entropy Group action Compact space 37B40 54H20 37B45 54F15 FOS: Mathematics Continuum (set theory) 0101 mathematics Mathematics - Dynamical Systems Mathematics |
| Zdroj: | Science China Mathematics |
| Popis: | Let $X$ be a compact metric space and $T:X\longrightarrow X$ be continuous. Let $h^*(T)$ be the supremum of topological sequence entropies of $T$ over all subsequences of $\mathbb Z_+$ and $S(X)$ be the set of the values $h^*(T)$ for all continuous maps $T$ on $X$. It is known that $\{0\} \subseteq S(X)\subseteq \{0, \log 2, \log 3, \ldots\}\cup \{\infty\}$. Only three possibilities for $S(X)$ have been observed so far, namely $S(X)=\{0\}$, $S(X)=\{0,\log2, \infty\}$ and $S(X)=\{0, \log 2, \log 3, \ldots\}\cup \{\infty\}$. In this paper we completely solve the problem of finding all possibilities for $S(X)$ by showing that in fact for every set $\{0\} \subseteq A \subseteq \{0, \log 2, \log 3, \ldots\}\cup \{\infty\}$ there exists a one-dimensional continuum $X_A$ with $S(X_A) = A$. In the construction of $X_A$ we use Cook continua. This is apparently the first application of these very rigid continua in dynamics. We further show that the same result is true if one considers only homeomorphisms rather than con\-ti\-nuous maps. The problem for group actions is also addressed. For some class of group actions (by homeomorphisms) we provide an analogous result, but in full generality this problem remains open. The result works also for an analogous class of semigroup actions (by continuous maps). 90 pages, the paper has been accepted for publication in SCIENCE CHINA Mathematics |
| Databáze: | OpenAIRE |
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