A simple characterization of the set ofμ-entropy pairs and applications
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We present simple characterizations of the setsE μ andE X of measure entropy pairs and topological entropy pairs of a topological dynamical system (X, T) with invariant probability measureμ. This characterization is used to show that the set of (measure) entropy pairs of a product system coincides with the product of the sets of (measure) entropy pairs of the component systems; in particular it follows that the product of u.p.e. systems (topological K-systems) is also u.p.e. Another application is to show that the proximal relationP forms a residual subset of the setE X . Finally an example of a minimal point distal dynamical system is constructed for whichE X ∩(X 0×X 0)≠\(\not 0\), whereX 0 is the denseG δ subset of distal points inX.
KeywordsDistal Point Topological Entropy Invariant Probability Measure Positive Entropy Topological Dynamical System
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