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

Cortez, María Isabel; Rivera-Letelier, Juan

Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 1, pp. 95-115.

Une conséquence bien connue du théorème de décomposition ergodique est que l'espace des mesures de probabilité invariantes d'un système dynamique topologique est un simplexe de Choquet métrisable et non vide. On montre que tout simplexe de Choquet métrisable et non vide se réalise comme l'espace des mesures de probabilité invariantes sur l'ensemble post-critique d'une application logistique. Ici, l'ensemble post-critique d'une application logistique est l'ensemble ω-limite de son unique point critique. En effet, on démontre que l'application logistique f peut être choisie de telle façon que son ensemble post-critique soit un ensemble de Cantor où f est minimal, et tel que chaque mesure de probabilité invariante sur cet ensemble soit d'exposant de Lyapunov null, et un état d'équilibre pour le potentiel $-\ln |f^{\text{'}}|$.

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 $-\ln |f^{\text{'}}|$.

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