0 D ec 1 99 8 Complete invariant for two - dimentional

We consider general Morse-Smale diffeomorphisms on a closed orientable two-dimentional surface. In this paper it is proved that the complete topological invariant of Morse-Smale diffeomorphisms is finite, the algorithm of the construction of the complete topological invariant in explicit form is given and necessary and sufficient conditions of topological conjugacy of Morse-Smale diffeomorphisms is obtained. This paper is devoted to the problem of the topological classification of Morse-Smale diffeomorphisms on two-dimensional manifolds. The basic difficulty here is the presence of a complicated structure of intersection of stable and unstable manifolds of periodic points, generating in the general case an infinite number of periodic trajectories. Nowadays this problem has rather large literature (see [2] – [16]). In particular, A. Z. Grines solved the problem in the paper [11] for the case of a finite number of heteroclinic trajectories and the topological classification of periodic components of diffeomorphisms is already given by A. Bezen in paper [2]. Other special cases were investigated in papers [7]–[10]. This problem in general case was considered in papers [12], [13], but this attempt was not crowned with success. This paper presents a complete finite topological invariant of MorseSmale diffeomorphisms and necessary and sufficient conditions of topological conjugacy of Morse-Smale diffeomorphisms. The construction is based on the local structure of direct product of the set of periodic and heteroclinic points. As applied to the structure of the intersection of stable and unstable manifolds of saddle periodic points it is called the lattice structure here. Structure of direct product generates relations on the set of heteroclinic points which in the case of Morse-Smale diffeomorphisms in turn lead to the finiteness of the complete topological invariant. This paper is improved and translated into English version of two articles to appear in Russian. There are many proofs improved in comparison with the basic variant. The definition of lattice neighborhood is introduced