A sufficient condition for the subordination principle in ergodic optimization

Let T : X → X be a continuous surjection of a topological space, and let f : X → ℝ be upper semi‐continuous. We wish to identify those T ‐invariant measures μ which maximize ∫ f d μ. We call such measures f ‐ maximizing , and denote the maximum by β ( f ). The study of such measures and their properties has recently been dubbed ergodic optimization . A first step to understanding the structure of a function's maximizing measures is to establish the following subordination principle defined by T. Bousch: if μ and ν are T ‐invariant measures such that supp ν ⊆ supp μ and μ is f ‐maximizing, then ν is also f ‐maximizing. Previous authors have approached this result by constructing a continuous function g : X → ℝ such that f − β( f ) ⩽ g ○ T − g . We provide a sufficient condition for the subordination principle which has advantages when the space X is noncompact.