June  2006, 16(2): 383-392. doi: 10.3934/dcds.2006.16.383

Every ergodic measure is uniquely maximizing


School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS

Received  April 2005 Revised  November 2005 Published  July 2006

Let $M_{\phi}$ denote the set of Borel probability measures invariant under a topological action $\phi$ on a compact metrizable space $X$. For a continuous function $f:X\to\R$, a measure $\mu\in\M_{\phi}$ is called $f$-maximizing if $\int f\, d\mu = s u p\{\int f dm:m\in\M_{\phi}\}$. It is shown that if $\mu$ is any ergodic measure in $\M_{\phi}$, then there exists a continuous function whose unique maximizing measure is $\mu$. More generally, if $\mathcal E$ is a non-empty collection of ergodic measures which is weak$^*$ closed as a subset of $\M_{\phi}$, then there exists a continuous function whose set of maximizing measures is precisely the closed convex hull of $\mathcal E$. If moreover $\phi$ has the property that its entropy map is upper semi-continuous, then there exists a continuous function whose set of equilibrium states is precisely the closed convex hull of $\mathcal E$.
Citation: Oliver Jenkinson. Every ergodic measure is uniquely maximizing. Discrete & Continuous Dynamical Systems, 2006, 16 (2) : 383-392. doi: 10.3934/dcds.2006.16.383

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