This note is a short exposition of the author’s lecture at the conference “Kolmogorov and con- temporary mathematics” (Moscow, 2003). The study of stochastic properties of smooth dynami-cal systems often leads to the investigation of stable attracting limit sets of a complex geometric nature, the so-called ‘strange attractors’. A problem in the topological classification of hyperbolic strange attractors is investigated.

This paper consists of two parts. The first, which is devoted to presenting results of Barge and Watkins, connects the closure of the union of the unstable manifolds of certain 'Smale horseshoes' with Knaster continua and projections on them of Vietoris-van Dantzig solenoids. In the second part the homeomorphism problem for expanding attractors of codimension 1 is solved when the dimension of the manifold generating the dynamical system is greater than two.

In the study of the phenomenon of strange attractors, that is, attracting sets of smooth dynamical systems having a complicated topological structure, essential use is made of properties such as hyperbolicity, which are observed in all known examples. However, for systems with discrete time the genuine hyperbolicity of strange attractors of codimension 1 imposes greater limitations on the topological structure of the manifold of the system than can be explained by the fact that in the analysis of specific systems the phenomenon of strict hyperbolicity is not observed. The aim of this article is to give a description of the topological and dynamical structure of hyperbolic orientable attractors of codimension 1 for diffeomorphisms of manifolds. We begin with some definitions. Let /: Μ -+M be a diffeomorphism of a smooth manifold Μ of dimension at least 2. An invariant hyperbolic set Λ is called an expanding attractor (or shrinking repellor) of codimension 1 if W (x) C Λ for any χ Ε Λ and dim W" (x) = η 1 (or if W(x) C Λ for χ G Λ and dim W(x) = η 1). In what follows we limit the discussion to expanding attractors of codimensional 1. An attractor Λ is called orientable if for any x, y Ε Λ and local invariant manifolds W\oc(x), W?oc(y) their intersection index is the same at all common points. If G is a connected open domain of Μ and xQ lies on the boundary 9G of the domain, then x 0 is said to be an accessible point of the boundary if there is an open arc in G one end of which is x0. The subset D{G) C dG consisting of all accessible points is called the accessible boundary of G. Let Λ be an attractor of a diffeomorphism f: M-*M and let G Μ \ Λ. The collection C, C, . . ., C of connected components of the accessible boundary D(M \ Λ) is called a connector of order m, or an m-connector, if they can be rearranged in such an order that C and C and also C and C' for 1 < / < m — 1 can be joined by open arcs of one-dimensional stable manifolds