Counterexamples to the open problem by Zhou and Feng on the minimal centre of attraction

• Published in: Nonlinearity 2012-05, Vol.25 (5), p.1443-1449
• Main Authors: OBADALOVA, Lenka, SMITAL, Jaroslav
• Format: Article
• Language: English
• Subjects: Attraction; Dynamical systems; Exact sciences and technology; General topology; Global analysis, analysis on manifolds; Invariants; Mathematical methods in physics; Mathematics; Metric space; Nonlinearity; Other topics in mathematical methods in physics; Physics; Probability and statistics; Probability theory and stochastic processes; Probability theory on algebraic and topological structures; Sciences and techniques of general use; Texts; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
• ISSN: 0951-7715; 1361-6544
• DOI: 10.1088/0951-7715/25/5/1443

Let (X, [functionof]) be a topological dynamical system, where X is a compact metric space and [functionof] : X arrow right X is a continuous map. Denote by M sub(x) the set of all invariant probability measures of [functionof] which are limit points of the sequence (ProQuest: formulae and/or non-USASCII text omitted), where delta sub(x) is the atomic probability measure on X with support {x}. We give a characterization of points x such that M sub(x) contains a measure whose support is C sub(x), the minimal centre of attraction of x, and provide examples showing that the characterization is nontrivial. In particular, the standard shift on two symbols, ([summationoperator] sub(2), [sigma]), contains a quasi-weakly almost periodic points y, z which are not weakly almost periodic such that C sub(y) is the support of an invariant measure and C sub(z) is not the support of an invariant measure. This solves in negative the Open Problem 4 raised by Zhou and Feng (2004 Nonlinearity 17 493-502).