Journal
QUALITATIVE THEORY OF DYNAMICAL SYSTEMS
Volume 22, Issue 3, Pages -Publisher
SPRINGER BASEL AG
DOI: 10.1007/s12346-023-00809-9
Keywords
Morse-Smale diffeomorphisms; Nielsen-Thurston theory; Homotopy classes; Homotopy types
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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.
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.
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