Extensions of dynamical systems without increasing the entropy

Let X be a compact metric space. For a large class of dynamical properties Λ (we call them hypertransitive properties) we prove that any continuous non-minimal selfmap f of X with property Λ can be extended to a triangular, i.e. skew product, map F(x, y) = (f(x), g(x, y)) in the space X × [0, 1] in such a way that F has the property Λ and all the fibre maps of F are monotone (hence the topological entropy of F is the same as that of f). For topological transitivity such an extension theorem was proved by Alsedà, Kolyada, Llibre and Snoha in 1999. Our result is much more general—in fact, the family of hypertransitive properties includes, except of transitivity, also total transitivity, weak mixing, χ-Banach transitivity and many other dynamical properties. Further we prove a similar extension theorem also for strong mixing. Moreover, both for hypertransitive Λ and for strong mixing, if we additionally assume that f has dense periodic points then we can guarantee the existence of an extension F which has all the required properties and also dense periodic points. Finally we show that for maps which are topologically exact (i.e. locally eventually onto) an analogous theorem does not work. However, we give an example showing that at least some of the topologically exact maps can be extended to triangular topologically exact maps with monotone fibres.