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 A_u and A_s acting on the space L₂=L₂ (X, F, μ) be “eigenfunctions” of the automorphism T with eigenvaluesθ _u > 1 andθ _s < 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 δ-fields chosen from these filtrations. Hyperbolic toral automorphisms are considered as an example. Applications to limit theorems are given. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 244, 1997, pp. 61–72. Translated by M. I. Gordin.