ar X iv : m at h / 04 07 51 7 v 2 [ m at h . C A ] 3 0 Ju l 2 00 4 OPERATORS , MARTINGALES , AND MEASURES ON PROJECTIVE LIMIT SPACES

Let X be a compact Hausdorff space. We study finite-to-one map-pings r : X → X, onto X, and measures on the corresponding projective limit space X∞(r). We show that the invariant measures on X∞(r) correspond in a one-to-one fashion to measures on X which satisfy two identities. Moreover, we identify those special measures on X∞(r) which are associated via our correspondence with a function V on X, a Ruelle transfer operator R V , and an equilibrium measure µ V on X.