Double extensions of dynamical systems and a construction of mixing filtrations.~II. Quasihyperbolic

M. I. Gordin

Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 3, Tome 260 (1999), pp. 103-118

Résumé

Let $T$ be an automorphism (an invertible measure preserving transformation) of a probability space $(X,\mathscr F,\mu)$ and let $U$ be a unitary operator on $L_2(X)=L_2(X,\mathscr F,\mu)$ defined by $Uf=f\circ T$. Let $A_s$ and $A_u$ be generators of symmetric Markov transition semigroups acting on $L_2$. $A_s$ and $A_u$ are supposed to satisfy the relations $$ U^{-1} A_s U=\theta^{-1} A_s,U^{-1} A_u U=\theta A_u $$ for some $\theta >1$. A nonnegative selfadjoint operator $A$ on $L_2$ with the properties $ UA=AU$, $ A_u+A_s\ge A$ is said to be a $T$-invariant minorant for $(A_u, A_s)$. Supposing that $A_u$ and $A_s$ commute, certain assumptions on a function $f \in L_2$ in terms of such a minorant are proposed under which the sequence $(f\circ T^k,k\in\mathbb Z)$ satisfies the functional form of the Central Limit Theorem and the Law of the Iterated Logarithm. A special case of these assumptions was considered in an earlier paper by the author. Quasihyperbolic toral automorphisms are considered as an application.