Ergodic optimization for generic continuous functions (Q968754)

Given a continuous transformation \(T: X \rightarrow X\) of a compact metric space, and an observable \(f: X \rightarrow \mathbb R\), the maximum ergodic average \(\beta(f)\) is given by the supremum over the integral of \(f\) with respect to the invariant measures of \(T\). The set of maximizing measures is denoted \(\mathcal{M}_{max}(f)\). The main result of this note is that if \(\mathcal{U}\) is an open dense subset of the closure of the set of \(T\)-ergodic measures, then the set of continuous observables \(f\) for which \(\mathcal{M}_{max}(f)\) intersected with this closure is contained in \(\mathcal{U}\) is open and dense in \(C(X)\). Conversely, given an open dense set \(U\) in \(C(X)\), the set of ergodic \(T\)-invariant measures which are maximizing for some \(f \in U\) is an open dense set in the set of ergodic \(T\)-invariant measures.