Chaos in countable dynamical system
Let \((X, \tau )\) be a countable compact Hausdorff space. For any \(Y\subset X\) let \(L(Y)\) be the set obtained from \(Y\) by removing the isolated points. The main result is the following characterization of spaces, for which there exists a distributionally chaotic continuous map \(F\) of \(X\) (for the notion of distributional chaos, cf. \textit{B. Schweizer} and the reviewer [Trans. Am. Math. Soc. 344, 737--754, (1994; Zbl 0812.58062)]): There is a \(\tau\)-metric \(\rho\) on \(X\) and a distributionally chaotic continuous map \(F: X\rightarrow X\) if and only if \(\#L^2(X)>1\). Here distributional chaos means existence of a distributionally chaotic pair of points. Moreover, if \(\#L^2(X)>1\) then there is a homeomorphism of \(X\) which is distributionally chaotic with respect to a suitable \(\tau\)-metric. The paper contains other similar interesting results.
scientific article; zbMATH DE number 1891208