The set of sequence entropies for graph maps (Q624424)

Let \((X, f)\) be a topological dynamical system, i.e., \(X\) is a compact metric space and \(f: X\to X\) is a continuous map. Denote by \(S(X)\) the set of all \(h^*(f)\) (maximal pattern entropies, which is the supremum of the sequence entropies over all subsequences of \(\mathbb{N}\)), where \(f\) runs over all continuous maps on \(X\). It was shown by \textit{J. S. Cánovas} [Nonlinearity 17, No. 1, 49--56 (2004; Zbl 1083.37036)] that if \(X= [0,1]\), then \(S(X)= \{\infty,0,\log 2\}\). The author, \textit{X. Ye} and \textit{R. Zhang} proved in [Nonlinearity 23, No. 1, 159--178 (2010; Zbl 1183.37031)] the same result when \(X\) is a finite tree. In this paper, the same conclusion is derived when \(X\) is a finite graph (non-empty connected compact one-dimensional polyhedron in \(\mathbb{R}^3\)).