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

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... 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.