The new approach to the topological classification of structurally stable diffeomorphisms and its realization for Morse-Smale diffeomorphisms on 2-manifold
by
Igor Vlasenko
Institute of Mathematics, NAS of Ukraine

Structurally stable diffeomorphisms are diffeomorphisms possessing a neighborhood in a space of diffeomorphisms in which all diffeomorphisms are topologically conjugate to them. They coincide (at least on 2-manifolds) with the set of diffeomorphisms whose periodic points are everywhere dense in the set of non-wandering points, the set of non-wandering points is hyperbolic, stable and unstable manifolds of non-wandering points intersect one another transversally.

In author's papers [1] and [2] there was given the topological classification of Morse-Smale diffeomorphisms in the dimension 2 which are structurally stable diffeomorphisms with finit sets of non-wandering points. It became possible because of the fact that the set of non-wandering and heteroclinic points being a locally maximum hyperbolic set possesses a local structure of direct product. In these papers there was introduced a method of construction of topological invariants of diffeomorphisms using induction on relations that appear due to the structure of direct product.

Since possessing of a local structure of direct product is a native property of structurally stable diffeomorphisms the method is also valid in general case. Its implementation may give the topological classification of structurally stable diffeomorphisms in the dimension 2. In higher dimensions using of this method allows to reduce the task of the topological classificatoin to some simpler task which seems to be solvable in the dimension 3.

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Igor Vlasenko  Complete invariant for two-dimensional Morse-Smale diffeomorfisms.  // e-Print, math.DS/9812063

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Igor Vlasenko  On the relations on the set of heteroclinic points of Morse-Smale diffeomorphisms on 2-manifolds.  // Methods of functional analysis and topology, to appear.

Date received: June 29, 1999