Given a continuous map T : X → X on a compact metric space X, we consider the push forward map T♯ : P(X) → P(X), and analyze how the dynamics of T relates to the dynamics of T♯ on the space of prob measures P(X).

The study of the dynamics of T♯ have been considered before, in a work by W. Bauer and K. Sigmund . They show, for example, that the topological entropy can be bounded below by the entropy of the map T and, if it is positive, then the topological entropy on the probabilistic setting is indeed infinity. More recently, the analysis of the push forward map T♯ have called much attention, mainly because of its importance in transport theory.

Motivated by this, we start to analyze properties of the dynamical system (X, T ) that can be transferred to the dynamical system (P(X),T♯). More specifically, we try to describe the properties that are (or are not) common to both transformations.