Stroboscopical property in topological dynamics
Stroboscopical property (s.p.) refers to a question posed by M. Misiurewicz for a minimal action \((X,f)\) of a continuous map \(f\) on a compact metric space \(X\): Is it true that for every sequence \(A=(n_k)\) of natural numbers, every point \(z\) is an \((A,\omega)\)@-limit point for some \(x_z\in X\) (which means \(T^{n_{k_i}}x_z\to z\) for a subsequence \(n_{k_i}\) of \(A\))? In the paper it is shown that the well-known Sturmian flow does not have this property (which settles Misiurewicz's question). For dimension one it is proved that all transitive systems on graphs satisfy s.p. In dimension 2 the authors construct a minimal skew product on the two-torus, where s.p. fails. In a more general setting it is proved that all (not only minimal) distal systems have s.p. Surjective equicontinuous systems are shown to have a ``uniform stroboscopical property''. Strong stroboscopical property (with the set of points \(x_z\) dense, for each \(A\) and \(z\)) is proved equivalent to strong topological mixing (on \(X\) locally compact). An interesting open question posed in the paper is whether all minimal weakly mixing systems have s.p.
scientific article; zbMATH DE number 1917839
scientific article