Double extensions of dynamical systems and a construction of mixing filtrations

M. I. Gordin

Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 2, Tome 244 (1997), pp. 61-72

Résumé

Let $T$ be an automorphism of a probability space $(X,\mathscr F,P)$ and let $A_s$ and $A_u$ be generators of symmetric Markov transition semigroups on $X$. $A_s$ and $A_u$ are supposed also to be “eigenvectors” for $T$ with eigenvalues $\theta^{-1}$ and $\theta$ for some $\theta>1$. We give a probabilistic construction (based on $A_s$ and $A_u$) of an extension of the quadruple $(X,\mathscr F,P,T)$. This extension $(X',\mathscr F',P,T')$ is naturally supplied with decreasing and increasing filtrations. Under the assumptions that $A_s$ and $A_u$ commute and that their sum $A_s+A_u$ is bounded below apart from zero we establish very strong decay to zero of the maximal correlation coefficient between the $\sigma$-fields of these filtrations. As an application we prove the following assertion under the above conjectures. Let $f\in L_2$ has integral 0 with respect to $P$ and be such that $$ \sum_{k\ge 0}\bigl((|f|^2_2-|\mathbf P_s(\theta^{-k})f|^2_2)^{1/2}+(|f|^2_2-|\mathbf P_u(\theta^{-k})f|^2_2)^{1/2}\bigr)\infty. $$ Then the sequence $\{f\circ T^k, k\in\mathbb Z\}$ satisfies the Functional Central Limit Theorem. As an example we consider hyperbolic toral automorphisms.