Strange attractors. Topologic, geometric and algebraic aspects

This article is a review on research work of two authors on hyperbolic and Lorenz like strange attractors. In 1971 R. Plykin received a letter from two young scientists of Warwick University David Chillingworth and Anthony Manning with pointing out his errors in the article entitled "The topology of basic sets for Smale diffeomorphisms". The main error in that manuscript was the statement about nonexistence of one dimensional hyperbolic attractor of the diffeomorphism of two-sphere. The first part of this report corrects previous errors and carries information about geometry and topology of hyperbolic strange attractors. The second part of the report contains some results obtained by N. Klinshpont on the problem of topological classification of Lorenz type attractors and their generalizations.
An investigation of stochastic properties of differentiable dynamical systems often results in study other limiting formations, the isolation of which is a difficult problem which requires the mobilization of not only analytical and geometrical methods but also substantial computational resources. In the geometrical approach, which this investigation follows, manifolds are studied together with dynamical systems on them in which the diversity of structures of attractors occurs and the main difficulties are the classification problems.