Topology and Topological Sequence Entropy

TL;DR

This paper classifies all possible sets of topological sequence entropy values for continuous maps on one-dimensional continua, showing that any such set within the known bounds can be realized, and introduces Cook continua into dynamics.

Contribution

It completely characterizes the sets of topological sequence entropy values for continuous maps on one-dimensional continua, using Cook continua for the construction.

Findings

All subsets between {0} and the known upper bounds are realizable.

The same classification applies to homeomorphisms.

Results extend to certain group and semigroup actions.

Abstract

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).