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

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

2009-01-24

arxiv.org/abs/0901.3835v2

Mathematics

Dynamical Systems

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

Scientific paper

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 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 $- \ln |f'|$.

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