Double extensions of dynamical systems and the construction of mixing filtrations. II: Quasihyperbolic toral automorphisms.
type: Item
instance of: scholarly article
title: Double extensions of dynamical systems and the construction of mixing filtrations. II: Quasihyperbolic toral automorphisms.
zbMATH Open document ID: 1163.37300
DOI: 10.1023/A:1014573115086
published in: Journal of Mathematical Sciences (New York)
publication date: 2002-06-25T00:00:00Z
review text: 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).
Mathematics Subject Classification ID: 37A05
Mathematics Subject Classification ID: 28D15
Mathematics Subject Classification ID: 37A30
Mathematics Subject Classification ID: 37D99
zbMATH DE Number: 1759040
author: Mikhaĭl Iosifovich Gordin
MaRDI profile type: Publication
full work available at URL: a:1014573115086
OpenAlex ID: W74051133
description: scientific article; zbMATH DE number 1759040