Stroboscopical property in topological dynamics
type: Item
instance of: scholarly article
title: Stroboscopical property in topological dynamics
zbMATH Open document ID: 1031.37013
DOI: 10.1016/S0166-8641(02)00186-4
author: Ľubomír Snoha
author: Víctor Jiménez López
published in: Topology and its Applications
publication date: 2003-05-27T00:00:00Z
review text: 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.
reviewed by: T. Downarowicz
Mathematics Subject Classification ID: 37B05
Mathematics Subject Classification ID: 37E30
Mathematics Subject Classification ID: 37A25
Mathematics Subject Classification ID: 54H20
zbMATH DE Number: 1917839
zbMATH Keywords: Misiurewicz stroboscopical property
zbMATH Keywords: minimal system
zbMATH Keywords: distal system
zbMATH Keywords: topological mixing
zbMATH Keywords: torus homeomorphism
MaRDI profile type: Publication
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full work available at URL: s0166-8641(02)00186-4
OpenAlex ID: W1516832093
description: scientific article; zbMATH DE number 1917839