Scaling entropy and automorphisms with purely point spectrum

We study the dynamics of the metrics generated by measure preserving transformations. We consider a sequence of average metrics and define the corresponding sequence of $\epsilon$-entropies ({\it scaling sequence}) of the measure with respect to the mean metrics. The main result claims that scaling sequences of an automorphism with respect to any {\it admissible metric} is bounded if and only if the automorphism has discrete spectrum. This gives a non-spectral criterion of the discreteness of the spectrum of an automorphism. The related result was discussed in \cite{Fe} but our approach is different. This article is one in the series of papers about asymptotic theory of sequences of the metric compacts with measure and its role in dynamics.Comment: 27 pp.28 Ref; St.Petersburg Mathematical Journal, No.1, 201

Duality and free measures in vector spaces, the spectral theory of actions of non-locally compact groups

The paper presents a general duality theory for vector measure spaces taking its origin in the author's papers written in the 1960s. The main result establishes a direct correspondence between the geometry of a measure in a vector space and the properties of the space of measurable linear functionals on this space regarded as closed subspaces of an abstract space of measurable functions. An example of useful new features of this theory is the notion of a free measure and its applications.Comment: 20 pp.23 Re

What does a generic Markov operator look like

We consider generic i.e., forming an everywhere dense massive subset classes of Markov operators in the space $L^2(X,\mu)$ with a finite continuous measure. Since there is a canonical correspondence that associates with each Markov operator a multivalued measure-preserving transformation (i.e., a polymorphism), as well as a stationary Markov chain, we can also speak about generic polymorphisms and generic Markov chains. The most important and inexpected generic properties of Markov operators (or Markov chains or polymorphisms) is nonmixing and totally nondeterministicity. It was not known even existence of such Markov operators (the first example due to M.Rozenblatt). We suppose that this class coinsided with the class of special random perturbations of $K$-automorphisms. This theory is measure theoretic counterpart of the theory of nonselfadjoint contractions and its application.Comment: 13 p.,Ref.1

On classification of measurable functions of several variables

We define a normal form (called the canonical image) of an arbitrary measurable function of several variables with respect to a natural group of transformations; describe a new complete system of invariants of such a function (the system of joint distributions); and relate these notions to the matrix distribution, another invariant of measurable functions found earlier, which is a random matrix. Bibliography: 7 titles.Comment: 17 p; J.Math.Sci.V.190 #3 (2013

Random and universal metric spaces

We introduce a model of the set of all Polish (=separable complete metric) spaces: the cone $\cal R$ of distance matrices, and consider geometric and probabilistic problems connected with this object. The notion of the universal distance matrix is defined and we proved that the set of such matrices is everywhere dense $G_{\delta}$ set in weak topology in the cone $\cal R$. Universality of distance matrix is the necessary and sufficient condition on the distance matrix of the countable everywhere dense set of so called universal Urysohn space which he had defined in 1924 in his last paper. This means that Urysohn space is generic in the set of all Polish spaces. Then we consider metric spaces with measures (metric triples) and define a complete invariant: its - matrix distribution. We give an intrinsic characterization of the set of matrix distributions, and using the ergodic theorem, give a new proof of Gromov's ``reconstruction theorem'. A natural construction of a wide class of measures on the cone $\cal R$ is given and for these we show that {\it with probability one a random Polish space is again the Urysohn space}. There is a close connection between these questions, metric classification of measurable functions of several arguments, and classification of the actions of the infinite symmetric group [V1,V2].Comment: 30 pages, substantially revised versio

Graded Lie Algebras and dynamical systems

The general class of the graded Lie algebras is defined. These algebras could be constructed using an arbitrary dynamical systems with discrete time and with invarinat measure. In this papers we consider the case of the central extension of Lie algebras which corresponds to the ordinary crossed product (as associative algebra) - series A. The structure of those Lie algebras is similar to Kac-Moody algebras, and these are a special case of so called algebras with continuous root system which were introduced by author with M.Saveliev in 90-th. The central extension open a new possibilty in algebraic theory of dynamical systems. The simpliest example corresponds to rotation of the circle (sine-algebra="quantum torus") and to adding of unity the additvie group of the p-adic integers.Comment: 10 pages, Late

Stable states and representations of the infnite symmetric group

We introduce the notion of stable representations, -- it is a new class of the representations of a certain class of groups which defined with positive definite functions which generalize the classical notion of the characters (or trace). We give the complete description of this class for infinite symmetric group ${\frak S}_{\Bbb N}$. It happened the family of the stable representations of ${\frak S}_{\Bbb N}$ coincides with the set of representations of the components (left or right) of the admissible representations of the double-group in the sense of Ol'shanski. In particular all these are representation of type ${\rm II}$ which was open question for many years.Comment: 6 pp. Ref.

Towards the definition of metric hyperbolicity

We introduce measure-theoretic definitions of {\it hyperbolic structure for measure-preserving automorphisms}. A wide class of $K$-automorphisms possesses a hyperbolic structure; we prove that all $K$-automorphisms have a slightly weaker structure of {\it semi-hyperbolicity}. Instead of the notions of stable and unstable foliations and other notions from smooth theory, we use the tools of the theory of polymorphisms. The central role is played by {\it polymorphisms} associated with a special invariant equivalence relation, more exactly, with a homoclinic equivalence relation. We call an automorphism with given hyperbolic structure a hyperbolic automorphism and prove that it is canonically quasi-similar to a so-called prime nonmixing polymorphism. We present a short but necessary vocabulary of polymorphisms and Markov operators from \cite{V1,V2}.Comment: 23 pp. Bibl. 1

Gel'fand-Tsetlin algebras, expectations, inverse limits, Fourier analysis

We define a general notion of Gel'fand-Tsetlin algebras in the framework of inductive limit of the family semisimple star-algebras and the notion of genralized conditional expectations. The inmverse limit of such family with respect to generalized expectation is bimodule over inductive limit. The most important examples conserned to inductive families of group algebras of classical series of the discrete and continuous groups. I start from some remarks about distinguished role of I.M.Gel'fand's conceptions in whole area.Comment: 18

Combinatorial encoding of continious dynamics, and transfer of the space of paths of the graded graphs

These notes follow my articles [1, 6], and give some new important details. We propose here a new combinatorial method of encoding of measure spaces with measure preserving transformations, (or groups of transformations) in order to give new, mostly locally finite geometrical models for investigation of dynamical properties of these objects.Comment: 12 pp. Ref.7 pic.2. arXiv admin note: text overlap with arXiv:1904.0292