Anatolii Mikhailovich Stepin

Theory of dynamical systems and general transformation groups with invariant measure

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.

Connections between the approximative and spectral properties of metric automorphisms

To each automorphism T of a Lebesgue space (X, Μ@#@) there corresponds a unitary operator UT in the space L2(X,Μ), defined by the formula (UTf) (x) = f (Tx),f ∃ L2(X,Μ), x ∃ X. In this note we investigate the special properties of the operator UT as a function of the rate of approximation of the automorphism T by periodic transformations (for the definition of the rate of approximation of a metric automorphism see [1]).

On asymptotic contractions

A constructive generalization of the contraction mapping principle to the case of mappings with arbitrary rate of contraction is given.

Isospectrality and Galois Projective Geometries

We construct a series of pairs of domains in the plane and pairs of surfaces with boundary that are isospectral but not isometric. The construction is based on the existence of finite transformation groups that are spectrally equivalent but not isomorphic.