Equicontinuity, Shadowing and Distality in General Topological Spaces
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Czechoslovak Mathematical Journal
16 pp
Online first
EQUICONTINUITY, SHADOWING AND DISTALITY IN GENERAL TOPOLOGICAL SPACES Huoyun Wang, Guangzhou Received November 4, 2018. Published online January 16, 2020.
Abstract. We consider the notions of equicontinuity point, sensitivity point and so on from a topological point of view. Many of these notions can be sensibly defined either in terms of (finite) open covers or uniformities. We show that for the notions of equicontinuity point and sensitivity point, Hausdorff or uniform versions coincide in compact Hausdorff spaces and are equivalent to the standard definitions stated in terms of a metric in compact metric spaces. We prove that a uniformly chain transitive map with uniform shadowing property on a compact Hausdorff uniform space is either uniformly equicontinuous or it has no uniform equicontinuity points. Keywords: shadowing; chain transitive; equicontinuity; uniform space MSC 2010 : 37B20, 37B05, 54H20
1. Introduction Throughout this paper (X, T ) denotes a topological dynamical system, where X is a Hausdorff space and T : X → X is a continuous map. A number of properties of interest in such systems are defined in purely topological terms, for example transitivity, minimality and proximality. Others are defined in terms of the particular metric, for example sensitivity, equicontinuity, chain transitivity and shadowing. When the phase space X is a compact metric space, the sensitivity and equicontinuity were studied, see [4], [13]. For example, the well-known Auslander-Yorke dichotomy theorem stated that a minimal dynamical system is either sensitive or all points are equicontinuous (see [4] and also [13]), which was further refined in [1]: a transitive system is either sensitive or has a dense set of equicontinuity points; it Supported by National Nature Science Funds of China (11771149, 11471125). DOI: 10.21136/CMJ.2020.0488-18
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was shown that if a chain transitive system has shadowing property then it is either sensitive or equicontinuous, see [10], Corollary 2.4. However, in fact it shows that many dynamical properties can be defined in a natural way on Hausdorff (but not necessarily compact or metric) spaces. It turns out that there are two sensible ways to do this, either in terms of finite open covers or in terms of uniformities (compatible with the topology). The uniform approach has been studied in a number of cases: Hood in [16] defined entropy for uniform spaces; Devaney chaos for uniform spaces was considered in [8] (for group actions); Auslander, Greschonig and Nagar in [3] generalized many known results about equicontinuity to the uniform spaces; P. Das and D. Das in [9] extended some results about shadowing to the uniform spaces, see also [14]. Recently, the open cover approach has been studied in a number of cases: Brian in [7] considered chain transitivity in terms of finite open covers in compact Hausdorff spaces; Good and Macias in [14] considered sensitivity and shadowing in terms of finite open covers in compact Hausdorff spaces. The present work is i
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