Choquet simplices as spaces of invariant probability measures of 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${}^{\ast }$ 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 on 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 on this set has zero Lyapunov exponent, and is an equilibrium state for the potential $-\ln |f^{\prime }|$.

34 pages. Minor changes. Accepted for publication in the Annales de l'Institut Henri Poincare, Analyse non lineaire

References in corpus (1)

• Invariant measures of minimal post-critical sets of logistic maps