Abstract
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.
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References
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.
F. Blanchard,A disjointness theorem involving topological entropy, Bulletin de la Société Mathématique de France121 (1993), 465–478.
F. Blanchard and Y. Lacroix,Zero-entropy factors of topological flows, Proceedings of the American Mathematical Society119 (1993), 985–992.
F. Blanchard, B. Host, A. Mass, S. Martínez and D. Rudolph,Entropy pairs for a measure, Ergodic Theory and Dynamical Systems15 (1995), 621–632.
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.
R. Ellis,The Veech structure theorem, Transactions of the American Mathematical Society186 (1973), 203–218.
S. Glasner,Proximal flows, Lecture Notes in Mathematics517, Springer-Verlag, Berlin, 1976.
E. Glasner and B. Weiss,On the construction of minimal skew products, Israel Journal of Mathematics34 (1979), 321–336.
E. Glasner and B. Weiss,Strictly ergodic uniform positive entropy models, Bulletin de la Société Mathématique de France122 (1994), 399–412.
E. Glasner and B. Weiss,Topological entropy of extensions, inErgodic Theory and its Connections with Harmonic Analysis, Cambridge University Press, 1995, pp. 299–307.
A. del Junco,On minimal self-joinings in topological dynamics, Ergodic Theory and Dynamical Systems7 (1987), 211–227.
W. Parry,Topics in Ergodic Theory, Cambridge University Press, 1981.
B. Weiss,Strictly ergodic models for dynamical systems, Bulletin of the American Mathematical Society13 (1985), 143–146.
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Glasner, E. A simple characterization of the set ofμ-entropy pairs and applications. Isr. J. Math. 102, 13–27 (1997). https://doi.org/10.1007/BF02773793
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DOI: https://doi.org/10.1007/BF02773793