Topological sequence entropy of \(\omega\)-limit sets of interval maps. (Q5951623)

Topological entropy, denoted \(h(f)\), was first studied by \textit{R. L. Adler}, \textit{A. G. Konheim} and \textit{M. H. McAndrew} [Trans. Am. Math. Soc. 114, 309--319 (1965; Zbl 0127.13102)] for a continuous map \(f: X \to X\), where \(X\) is a compact topological space. For a metric space \(X\), \textit{R. Bowen} [Trans. Am. Math. Soc. 153, 401--414 (1971; Zbl 0212.29201)] provided an alternate definition of topological entropy for a uniformly continuous map \(f: X \to X\).NEWLINENEWLINEFor a compact metric space \(X\), \textit{T. N. T. Goodman} [Proc. Lond. Math. Soc., III. Ser. 29, 331--350 (1974; Zbl 0293.54043)] introduced the topological sequence entropy \(h_S(f)\) for a continuous map \(f: X \to X\). Let \(\omega(x,f)\) denote the omega limit set of \(x \in X\) for \(f: X \to X\). For a continuous map \(f: [0,1] \to [0,1]\), \textit{A. Blokh} [in: Jones, Christopher K. R. T. (ed.) et al., Dynamics reported. Expositions in dynamical systems. New series. Vol. 4, Berlin, Springer, 1--59 (1995; Zbl 0828.58009)] proved \(h(f)= \sup_{x \in [0,1]} h(f| _{\omega(x,f)})\).NEWLINENEWLINEIn the paper under review, the author proves for a continuous map \(f: [0,1] \to [0,1]\) with zero topological entropy that \(h_S(f| _{\omega(x,f)})=0\) for any \(x \in [0,1]\) and any increasing sequence \(S\) of positive integers. Moreover, the author shows for \(f\) chaotic in the sense of \textit{T.-Y. Li} and \textit{J. A. Yorke} [Am. Math. Mon. 82, 985--992 (1975; Zbl 0351.92021)] with zero topological entropy that Blokh's formula is false if topological entropy is replaced by topological sequence entropy.