2009/01/24 by Cortez, Maria Isabel, Rivera-Letelier, Juan
#37E05 #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.0901.3835
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 ω-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'|.