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A characterization of \(\mu\)-equicontinuity for topological dynamical systems. (English) Zbl 1378.37015

A topological dynamical system \((X,f)\) is a continuous transformation \(f\) on a compact metric space \(X\). Let \((X,f)\) be a topological dynamical system and let \(x\in X\). We say that \(x\) is an equicontinuous point if for every \(y\) close to \(x\) we have that \(f^{i}(x)\) and \(f^{i}(y)\) stay close for all \(i\). Thus, a topological dynamical system is equicontinuous if and only if every point is an equicontinuous point.
Since the equicontinuity is a very strong property, for several years different attempts have been made to weaken this property. For example:
in [Ergodic Theory Dyn. Syst. 7, 105–118 (1987; Zbl 0588.68029) ] R. H. Gilman introduced the concept of \(\mu\)-equicontinuity points and \(\mu\)-sensitivity.
in [J. Math. Anal. Appl. 309, No. 1, 375–382 (2005; Zbl 1089.28011)] B. Cadre and P. Jacob introduced \(\mu\)-pairwise sensitivity.
In this paper the author defines a subclass of \(\mu\)-equicontinuous systems based on a local periodicity notion, and he provides another characterization of the notion of measure theoretical sensitivity.

MSC:

37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
37A50 Dynamical systems and their relations with probability theory and stochastic processes
54H20 Topological dynamics (MSC2010)
28A75 Length, area, volume, other geometric measure theory
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