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Mathematics, Dynamical Systems
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).
