A simple characterization of the set of μ-entropy pairs and applications

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 ∩ (X₀ × X₀) ≠ ∅, where X₀ is the dense G_δ subset of distal points in X.