Double extensions of dynamical systems and constructing mixing filtrations

Let T: X → X be an automorphism (a measurable invertible measure-preserving transformation) of a probability space (X, F, μ) and let two μ-symmetric Markov generators Au and As acting on the space L2 = L2(X, F, μ,) be " eigenfunctions" of the automorphism T with eigenvalues θu > 1 and θs < 1, respectively. We construct an extension of the automorphism T having increasing and decreasing filtrations by means of a transformation on the path space of these processes. Under additional conditions, we give an estimate of the maximal correlation coefficient between the σ-fields chosen from these filtrations. Hyperbolic toral automorphisms are considered as an example. Applications to limit theorems are given. ©2000 Kluwer Academic/Plenum Publishers.