3. The Topological Classification of the Gradient-Like Diffeomorphism on Surfaces

In this chapter we study the periodic data (the period, the Morse index, the orientation type) of the periodic orbits of gradient-like diffeomorphisms on orientable surfaces. Though such diffeomorphisms are similar in many ways to structurally stable flows on surfaces they have one property which makes them considerably different. This property is a possible non-trivial periodic action of the diffeomorphism in the fundamental group of the surface. The study of admissible collections of periodic data made it possible to solve the problem of realization of gradient-like diffeomorphisms. It also showed the interrelation between the dynamics of such diffeomorphisms and periodic transformations of surfaces whose classification is an important part of Nielsen-Thurston theory. In the present chapter we introduce a topological invariant for gradient-like diffeomorphisms on orientable surfaces. This invariant is a graph similar to that of Peixoto for structurally stable flows without cycles. We prove that such a graph equipped with a permutation of the set of the vertices completely determines the class of topological conjugacy of a gradient-like diffeomorphism on a surface. Moreover, we construct another complete topological invariant for these diffeomorphisms (a scheme) which is based on the representation of the dynamics of a diffeomorphism as “attractor-repeller” and on the subsequent study of the space of wandering orbits. We show that the class of topological conjugacy of a gradient-like diffeomorphism is determined (up to a conjugating homeomorphism) by a collection of 2-tori each of which has a family of circles embedded into it. The results on the topological classification of special classes of the Morse-Smale diffeomorphisms on 2-manifolds can be found in [1‐6, 8, 9].