Determination of the Homotopy Type of a Morse-Smale Diffeomorphism on an Orientable Surface by a Heteroclinic Intersection

This paper is devoted to the study of homotopy types of orientation-preserving Morse-Smale diffeomorphisms on closed orientable surfaces. Since any Morse-Smale diffeomorphism has a finite set of periodic points, then, according to the Nielsen-Thurston classification, it is homotopic to either a periodic homeomorphism or an algebraically finite order homeomorphism. It follows from the results of V. Grines and A. Bezdenezhnykh that any gradient-like diffeomorphism is homotopic to a periodic homeomorphism. However, when the wandering set of a given diffeomorphism contains heteroclinic intersections, then the question of its homotopy type is remains open. In the present work, an algorithm for recognizing the homotopy type of a non-gradient-like Morse-Smale diffeomorphism by its heteroclinic intersection is proposed. The algorithm is based on the construction of a filtration for a diffeomorphism and calculation of the intersection index of saddle separatrices in the fundamental annuli of filtration elements. It is established that a Morse-Smale diffeomorphism is homotopic to a periodic homeomorphism if and only if the total intersection index over all homotopic annuli is equal to zero.

Automated summary

This paper investigates the homotopy types of orientation-preserving Morse-Smale diffeomorphisms on closed orientable surfaces. It is established that any Morse-Smale diffeomorphism is homotopic to either a periodic homeomorphism or an algebraically finite order homeomorphism. The author proposes an algorithm for recognizing the homotopy type of a non-gradient-like Morse-Smale diffeomorphism based on its heteroclinic intersection. It is proven that a Morse-Smale diffeomorphism is homotopic to a periodic homeomorphism if and only if the total intersection index over all homotopic annuli is equal to zero.