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

TL;DR

This paper demonstrates that any non-empty metrizable Choquet simplex can be realized as the space of invariant probability measures on the post-critical set of a logistic map, linking dynamical systems and convex geometry.

Contribution

It constructs logistic maps whose post-critical sets are Cantor sets with minimal dynamics, realizing arbitrary Choquet simplices as invariant measure spaces.

Findings

Any non-empty metrizable Choquet simplex can be represented as an invariant measure space of a logistic map.

The constructed logistic maps have post-critical sets that are Cantor sets with minimal dynamics.

Invariant measures on these sets have zero Lyapunov exponent and are equilibrium states for a specific potential.

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 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 $- \ln |f'|$.