| 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. |
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