The theory of dynamical systems with invariant measure, or ergodic theory, is one of those domains of mathematics whose form changed radically in the last 15-20 years. This has to do both with the internal problems of ergodic theory End with its connections with other parts of mathematics. In ergodic theory itself, there arose the theory of entropy of dynamical systems, whose origin was in the papers of A. N. Kolmogorov. Recently, remarkable progress has been made by Ornstein and his collaborators in the problem of metric isomorphism of Bernoulli automorphisms and K-automorphisms, i.e., dynamical systems with very strong mixing properties. Another important event of recent times is a new, profound connection of ergodic theory with statistical mechanics, not only enriching ergodic theory itself, but also leading to new progress in the mathematical problems of statistical mechanics. Both of the circles of problems mentioned occupy a significant place in this survey. On the other hand, a series of applications of ergodic theory is intentionally excluded from our survey. This has to do in the first place with physics papers, which do not contain strictly mathematical results. Also, we shall not dwell on many mathematical papers connected in one way or another with ergodic theory, but not relating directly to it. Such, for example, are the papers of G. A. Margulis and Mostow on quasiconformal mappings of manifolds of negative curvature, in which the ergodicity of flows on such manifolds is used, or the papers of Glimm and Jaffe, which can be partially interpreted as investigations of the mixing properties of some dynamical systems which arise in quantum field theory.

Theory of dynamical systems and general transformation groups with invariant measure

A systematic survey of the theory of linear evolution equations in Banach spaces, reviewed in the period 1968–1982 in Ref Zh Matematika, is presented

Differential equations in a Banach space

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Ergodic theorems for contraction semi-groups

We show that the Maharam skew product ofμ-recurrent nonsingular endomorphisms is conservative and give some applications. Among them is the construction of a conservative ergodic invertible natural extension forμ-recurrent ergodic nonsingular endomorphisms.

Onμ-recurrent nonsingular endomorphisms

The article considers the task of determining the role of specialists who can be involved in assessing the degree of destruction of a railway object as a result of an emergency. The initial data of the task is the degree of achievement of a common goal by various subgroups (coalitions) of specialists, obtained from the results of training. The solution of this task is implemented using a fuzzy cooperative game with side payments, for it, we introduce the concept of a solution consisting of a single division and coinciding with the C-core (core), the Neumann-Morgenstern solution with a certain dominance relation, and the N-core (nucleolus) is introduced for it. For games with n allowed coalitions, we present an algorithm for finding the solution in question, which is illustrated by a meaningful example.

On the Concept of Solving a Fuzzy Cooperative Game with Side Payments

https://projecteuclid.org/download/pdf_1/euclid.tmj/1178230100

Vector valued ergodic theorems for a one-parameter semigroup of linear operators

On the recurrence theorems in ergodic theory

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Mathematical foundation of general cooperative fuzzy games

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On general ergodic theorems, II

1, The purpose of the present paper is to extend a general ergodic theorem [1] and a general ergodic theorem of Abel type [4] in the discrete case to one in the continuous case. 2. Consider a 6-finite measure space (X, F, p) and also a measure space (R+, 51, dt) where R+= [0, cc), 51 is the 6-algebra of all Lebesgue measurable subsets of R+ and dt the Lebesgue measure on <511. Let L1 be the real or complex Banach space of all equivalence classes of real or complex valued integrable functions on X. Let {Tt : t e R~} be a strongly continuous semigroup of linear contractions on L1. Then it is known that, given f e L1, there exists a 510'-measurable function g on R+®X such that, for every t, g(t, x) _ (T t f)(x) for a. a. x. Such a function g is uniquely determined up to a set of dt®dp-measure zero. In what follows, g(t, x) will be denoted by (T t f)(x) . Then, by Fubini's theorem it is shown that, for ax  a. x

Shigeru Tsurumi

Papers

Vector valued ergodic theorems for a one-parameter semigroup of linear operators

On the recurrence theorems in ergodic theory

Mathematical foundation of general cooperative fuzzy games

On general ergodic theorems, II

An ergodic theorem for a semigroup of linear contractions