Ergodic optimization for noncompact dynamical systems
The purpose of this note is to initiate the study of ergodic optimization for general topological dynamical systems T:X X, where the topological space X need not be compact. Given four possible notions of largest ergodic average are defined; for compact metrisable X these notions coincide, while for general Polish spaces X they are related by inequalities, each of which may be strict. We seek conditions on f which guarantee the existence of a normal form, in order to characterize its maximizing measures in terms of their support. For compact metrisable X it suffices to find a fixed point form. For general Polish X this is not the case, but an extra condition on f, essential compactness, is shown to imply the existence of a normal form. When T:X X is the full shift on a countable alphabet, essential compactness yields an easily checkable criterion for the existence of a normal form.