Invariant measures of minimal post-critical sets of logistic maps

Date
Authors
Cortez, María Isabel
Rivera-Letelier, Juan
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
Description
We construct logistic maps whose restriction to the omega-limit set of its critical point is a minimal Cantor system having a prescribed number of distinct ergodic and invariant probability measures. In fact, we show that every metrizable Choquet simplex whose set of extreme points is compact and totally disconnected can be realized as the set of invariant probability measures of a minimal Cantor system corresponding to the restriction of a logistic map to the omega-limit set of its critical point. Furthermore, we show that such a logistic map $f$ can be taken so that each such invariant measure has zero Lyapunov exponent and is an equilibrium state of $f$ for the potential $-\ln |f'|$.
Comment: 30 pages, minor corrections, to appear in Israel J. Math
Keywords
Mathematics - Dynamical Systems, 37E05 (Primary), 37A99, 37B10, 54H20 (Secondary)
Citation
Collections