Choquet simplices as spaces of invariant probability measures on post-critical sets

Abstract

A well-known consequence of the ergodic decomposition theorem is that the space of invariant probability Measures of a topological dynamical system, endowed with the weak* topology, is a non-empty metrizable Choquet simplex. We show that every non-empty metrizable Choquet simplex arises as the space of invariant probability measures oil the post-critical set of a logistic map. Here. the post-critical set of a logistic map is the omega-limit set of its unique critical point. In fact we show the logistic map f can be taken in such a way that its post-critical set is a Cantor set where f is minimal, and Such that each invariant probability measure oil this set has zero Lyapunov exponent, and is,in equilibrium state for the potential - In vertical bar f'vertical bar. (C) 2009 Elsevier Masson SAS. All rights reserved.