Several remarks on Pascal automorphism and infinite ergodic theory

We interpret the Pascal-adic transformation as a generalized induced automorphism (over odometer) and formulate the $\sigma$-finite analog of odometer which is also known as "Hajian-Kakutani transformation" (former "Ohio state example"). We shortly suggest a sketch of the theory of random walks on the groups on the base of $\sigma$-finite ergodic theory.Comment: 14 pp,Ref.1

Disintegration of positive isometric group representations on Lp\mathrm{L}^p-spaces

Let $G$ be a Polish locally compact group acting on a Polish space $X$ with a $G$-invariant probability measure $\mu$. We factorize the integral with respect to $\mu$ in terms of the integrals with respect to the ergodic measures on $X$, and show that $\mathrm{L}^p(X,\mu)$ ($1\leq p<\infty$) is $G$-equivariantly isometrically lattice isomorphic to an $\mathrm{L}^p$-direct integral of the spaces $\mathrm{L}^{p}(X,\lambda)$, where $\lambda$ ranges over the ergodic measures on $X$. This yields a disintegration of the canonical representation of $G$ as isometric lattice automorphisms of $\mathrm{L}^p(X,\mu)$ as an $\mathrm{L}^p$-direct integral of order indecomposable representations. If $(X^\prime,\mu^\prime)$ is a probability space, and, for some $1\leq q<\infty$, $G$ acts in a strongly continuous manner on $\mathrm{L}^q(X^\prime,\mu^\prime)$ as isometric lattice automorphisms that leave the constants fixed, then $G$ acts on $\mathrm{L}^{p}(X^{\prime},\mu^{\prime})$ in a similar fashion for all $1\leq p<\infty$. Moreover, there exists an alternative model in which these representations originate from a continuous action of $G$ on a compact Hausdorff space. If $(X^\prime,\mu^\prime)$ is separable, the representation of $G$ on $\mathrm{L}^p(X^\prime,\mu^\prime)$ can then be disintegrated into order indecomposable representations. The notions of $\mathrm{L}^p$-direct integrals of Banach spaces and representations that are developed extend those in the literature.Comment: Section on future perspectives added. 35 pages. To appear in Positivit

The classification problem for automorphisms of C*-algebras

We present an overview of the recent developments in the study of the classification problem for automorphisms of C*-algebras from the perspective of Borel complexity theory.Comment: 21 page

Arithmetic Dynamics

This survey paper is aimed to describe a relatively new branch of symbolic dynamics which we call Arithmetic Dynamics. It deals with explicit arithmetic expansions of reals and vectors that have a "dynamical" sense. This means precisely that they (semi-) conjugate a given continuous (or measure-preserving) dynamical system and a symbolic one. The classes of dynamical systems and their codings considered in the paper involve: (1) Beta-expansions, i.e., the radix expansions in non-integer bases; (2) "Rotational" expansions which arise in the problem of encoding of irrational rotations of the circle; (3) Toral expansions which naturally appear in arithmetic symbolic codings of algebraic toral automorphisms (mostly hyperbolic). We study ergodic-theoretic and probabilistic properties of these expansions and their applications. Besides, in some cases we create "redundant" representations (those whose space of "digits" is a priori larger than necessary) and study their combinatorics.Comment: 45 pages in Latex + 3 figures in ep