Every ergodic measure is uniquely maximizing

Oliver Jenkinson
2006 Discrete and Continuous Dynamical Systems. Series A  
Let M φ denote the set of Borel probability measures invariant under a topological action φ on a compact metrizable space X. For a continuous function f : It is shown that if µ is any ergodic measure in M φ , then there exists a continuous function whose unique maximizing measure is µ. More generally, if E is a non-empty collection of ergodic measures which is weak * closed as a subset of M φ , then there exists a continuous function whose set of maximizing measures is precisely the closed
more » ... x hull of E. If moreover φ 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 E.
doi:10.3934/dcds.2006.16.383 fatcat:tjitybofnnhhbg2yb6sxzyulaq