Commutativity and non-commutativity of topological sequence entropy
Topological sequence entropy [as defined by \textit{T. N. T. Goodman}, Proc. Lond. Math. Soc. (3) 29, 331-350 (1974; Zbl 0293.54043)] is a natural extension of topological entropy [introduced by \textit{R. L. Adler, A. G. Konheim} and \textit{M. H. McAndrew}, Trans. Am. Math. Soc. 114, 309-319 (1965; Zbl 0127.13102) and in case of metrizable space by \textit{R. Bowen}, Trans. Am. Math. Soc. 153, 401-414 (1971; Zbl 0212.29201), \textit{E. I. Dinaburg}, Izv. Akad. Nauk SSSR, Ser. Mat. 35, 324-366 (1971; Zbl 0216.44704)]. One should reasonably expect that the commutativity property also holds for it. This paper is devoted to commutativity and noncommutativity properties of topological sequence entropy. First, the authors prove that if \(X\) is a compact metric space and \(f,g:X\to X\) are continuous maps then \(h_A(f\circ g)= h_A(g\circ f)\) for every increasing sequence \(A\) if \(X= [0,1]\). Then they construct a counterexample to the commutativity formula in the setting of an appropriate compact metric space using so-called weakly unimodal maps of type \(2^\infty\). In the interim, they also show that the equality \(h_A(f)= h_A(f|_{\bigcap_{n\geq 0}f^n(X)})\) is true if \(X= [0,1]\) but does not necessarily hold if \(X\) is an arbitrary compact metric space. Commutativity for continuous selfmaps of a compact space was first given by \textit{S. F. Kolyada} and \textit{L. Snoha} [Random Comput. Dyn. 4, 205-233 (1996; Zbl 0909.54012)].
scientific article; zbMATH DE number 1355997
scientific article