474 research outputs found
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 -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
Dynamics of metrics in measure spaces and their asymptotic invariants
We discuss the Kolmogorov's entropy and Sinai's definition of it; and then
define a deformation of the entropy, called {\it scaling entropy}; this is also
a metric invariant of the measure preserving actions of the group, which is
more powerful than the ordinary entropy. To define it, we involve the notion of
the -entropy of a metric in a measure space, also suggested by A. N.
Kolmogorov slightly earlier. We suggest to replace the techniques of measurable
partitions, conventional in entropy theory, by that of iterations of metrics or
semi-metrics. This leads us to the key idea of this paper which as we hope is
the answer on the old question: what is the natural context in which one should
consider the entropy of measure-preserving actions of groups? the same question
about its generalizations--scaling entropy, and more general problems of
ergodic theory.
Namely, we propose a certain research program, called {\it asymptotic
dynamics of metrics in a measure space}, in which, for instance, the
generalized entropy is understood as {\it the asymptotic Hausdorff dimension of
a sequence of metric spaces associated with dynamical system.} As may be
supposed, the metric isomorphism problem for dynamical systems as a whole also
gets a new geometric interpretation.Comment: 19 pp. Ref.2
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
Towards the definition of metric hyperbolicity
We introduce measure-theoretic definitions of {\it hyperbolic structure for
measure-preserving automorphisms}. A wide class of -automorphisms possesses
a hyperbolic structure; we prove that all -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
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 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 -automorphisms. This theory is measure theoretic
counterpart of the theory of nonselfadjoint contractions and its application.Comment: 13 p.,Ref.1
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
A new approach to the representation theory of the symmetric groups, III: Induced representations and the Frobenius--Young correspondence
We give a new (inductive) proof of the classical Frobenius--Young
correspondence between irreducible complex representations of the symmetric
group and Young diagrams, using the new approach, suggested in \cite{OV, VO},
to determining this correspondence. We also give linear relations between
Kostka numbers that follow from the decomposition of the restrictions of
induced representations to the previous symmetric subgroup. We consider a
realization of representations induced from Young subgroups in polylinear forms
and describe its relation to Specht modules.Comment: 22p. Ref.1
Random and universal metric spaces
We introduce a model of the set of all Polish (=separable complete metric)
spaces: the cone 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 set in weak topology in the cone .
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 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
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