### Article

## Существование связного характеристического пространства у градиентно-подобных диффеоморфизмов поверхностей

In this paper we consider the class GG orientation preserving gradient-like diffeomorphisms ff defined on a smooth oriented closed surface M2. Establishes that for any such pair of diffeomorphisms there is a dual attractor-repeller Af,Rf, which have a topological dimension not greater than 1 and the orbit space in their Supplement Vf (characteristic space) is homeomorphic to the two-dimensional torus. The immediate consequence of this result is, for example, the same period all of separatrices of a saddle of diffeomorphisms f∈G On the possibility of such representation of the dynamics of the system in the form `source-drain" founded a number of classification results for a structurally stable dynamical systems with dabloidami a set consisting of a finite number of orbits of systems of Morse-Smale. For example, for systems in dimension three, there is always a coherent characteristic space associated with the choice of a one-dimensional dual pairs of attractor-repeller. In dimension two this is not true even in the gradient-like case, however, the paper shows that there is a one-dimensional dual pair, the characteristic of the orbit space which is connected.