Invariant measures of minimal post-critical sets of logistic maps |
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Authors: | María Isabel Cortez Juan Rivera-Letelier |
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Institution: | (1) Department of Mathematics, College of Humanities and Sciences, Nihon University 3-25-40 Sakura Jyousui, Setagayaku Tokyo, 156-8550, Japan, |
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Abstract: | We construct logistic maps whose restriction to the ω-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 ω-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′|. |
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Keywords: | |
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