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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- [B,1]F. Blanchard,Fully positive topological entropy and topological mixing, inSymbolic Dynamics and Its Applications, Contemporary Mathematics, Vol. 135, American Mathematical Society, Providence, 1992, pp. 95–105.Google Scholar
- [BGH]F. Blanchard, E. Glasner and B. Host,A variation on the variational principle and applications to entropy pairs, Ergodic Theory and Dynamical Systems, to appear.Google Scholar
- [GW,3]E. Glasner and B. Weiss,Topological entropy of extensions, inErgodic Theory and its Connections with Harmonic Analysis, Cambridge University Press, 1995, pp. 299–307.Google Scholar
- [P]W. Parry,Topics in Ergodic Theory, Cambridge University Press, 1981.Google Scholar