Double extensions of dynamical systems and the construction of mixing filtrations. II: Quasihyperbolic toral automorphisms.
Let \(T\) be an automorphism (an invertible measure preserving transformation) of a probability space \((X,{\mathcal F}, \mu)\) and let \(U\) be a unitary operator on \(L_2 (X) =L_2 (X, {\mathcal F}, \mu)\) defined by \(Uf=f\circ T\). Let \(A_s\) and \(A_u\) be the generators of symmetric Markov transition semigroups acting on \(L_2\). It is assumed that \(A_s\) and \(A_u\) satisfy the relations \[ U^{-1} A_sU=\theta^{-1}A_s \quad\text{and}\quad U^{-1}A_u U=\theta A_u \] for some \(\theta>1\). A nonnegative self-adjoint operator \(A\) on \(L_2\) with the properties \(UA=AU\) and \(A_u+A_s\geq A\) is called a \(T\)-invariant minorant for \((A_u,A_s)\). Assuming 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. For Part I, see the authors, J. Math. Sci., New York 99, No.~2, 1053--1060 (2000); translation from Zap. Nauchn. Semin. POMI 244, 61--72 (1997; Zbl 0973.37004).
scientific article; zbMATH DE number 1759040