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A simple characterization of the set of \(\mu\)-entropy pairs and applications. (English) Zbl 0909.54035

Topological entropy pairs (resp. measure entropy pairs) were introduced by F. Blanchard [Bull. Soc. Math. Fr. 121, No. 4, 465-478 (1993; Zbl 0814.54027)] (resp. F. Blanchard, B. Host, A. Maass, S. Martinez and D. J. Rudolph [Ergodic Theory Dyn. Syst. 15, No. 4, 621-632 (1995; Zbl 0833.58022)]). In this note the author gives a simple characterization of the sets of these points. This leads (among other properties) to the result that the set of entropy pairs of a product transformation is the product of the sets of entropy pairs of their components (positive entropy is assumed).
This paper is a nice example for the interplay between measure theoretic and topological concepts. In the second part (using the measure theoretic concept), two questions, raised by J. Auslander, are answered: (1) It is shown that the set of topological entropy pairs for a point distal transformation is contained in \(X_0\times X_0\) where \(X_0\) is the set of distal points. (2) It is also true that in a general system of positive entropy the set of topological entropy pairs contains the proximal relation as a residual set.

MSC:

54H20 Topological dynamics (MSC2010)
28D20 Entropy and other invariants
54C70 Entropy in general topology
37A99 Ergodic theory
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References:

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