SPATIAL-TEMPORAL DIFFERENTIATION THEOREMS

Let (X,B,mu,T) be a dynamical system where X is a compact metric space with Borel sigma-algebra B, and mu is a probability measure that is ergodic with respect to the homeomorphism T: X -> X. We study the following differentiation problem: Given f is an element of C(X) and F-k is an element of B, where mu(F-k) > 0 and mu(F-k) -> 0, when can we say that (lim )(k ->infinity)integral(Fk)(1/k( )Sigma(k-1)(i=0)T(i)f)d mu/mu(F-k) = integral fd(mu) ? We establish some sufficient conditions for these sequences to converge.