A Bilateral Version of the Shannon-McMillan-Breiman Theorem

We give a new version of the Shannon-McMillan-Breiman theorem in the case of a bijective action. For a finite partition α of a compact set X and a measurable action T on X, we denote by CT n,m,α(x) the element of the partition α ∨ T 1α ∨ . . . ∨ Tmα ∨ T−1α ∨ . . . ∨ T−nα which contains a point x. We prove that for μ-almost all x, lim n+m→∞ ( −1 n+m ) logμ(C n,m,α(x)) = hμ(T, α), where μ is a T -ergodic probability measure and hμ(T, α) is the metric entropy of T with respect to the partition α.