## Abstract

We present simple characterizations of the sets E_{μ} and E_{x} of measure entropy pairs and topological entropy pairs of a topological dynamical system (X,T) with invariant probability measure μ. 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 relation P forms a residual subset of the set E_{X}. Finally an example of a minimal point distal dynamical system is constructed for which E_{X} ∩ (X_{0} × X_{0}) ≠ ∅, where X_{0} is the dense G_{δ} subset of distal points in X.

Original language | English |
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Pages (from-to) | 13-27 |

Number of pages | 15 |

Journal | Israel Journal of Mathematics |

Volume | 102 |

DOIs | |

State | Published - 1997 |