A disjointness theorem involving topological entropy

Abstract

— A cover of some compact set X by two non dense open sets is called a standard cover. In the cartesian square of a flow (X, T ), pairs (x, x′) outside the diagonal are defined as entropy pairs whenever any standard cover (U, V ) such that (x, x′) ∈ Int(Uc) × Int(V c) has positive entropy. The set of such pairs is nonempty provided h(X, T ) > 0 ; it is T × T -invariant, and all pairs in its closure belong either to it or to the diagonal. A flow is said to have uniform positive entropy if any standard cover has positive entropy (or if all non diagonal pairs are entropy pairs). Properties of entropy pairs are used to show that flows with uniform positive entropy (in fact a wider class) are disjoint from minimal flows with entropy 0. A flow with uniform positive entropy containing only one periodic orbit is constructed. (*) Texte recu le 10 mars 1992. F. BLANCHARD, Laboratoire de Mathematiques Discretes, case 930, 163 av. de Luminy, 13288 Marseille Cedex 09 (France). e-mail : blanchar@lmd.univ-mrs.fr. AMS classification : 54H20. BULLETIN DE LA SOCIETE MATHEMATIQUE DE FRANCE 0037–9484/1993/465/$ 5.00 c © Societe mathematique de France