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

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Abstract

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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Notes

  1. For Morse-Smale diffeomorphisms with a finite set of heteroclinic orbits given on a two-dimensional torus, the theorem was proved in [13].

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Acknowledgements

The work was supported by the Russian Science Foundation (Grant 21-11-00010), except Section 3, which was carried out on the basis of the International Laboratory of Dynamical Systems and Applications of the National Research University Higher School of Economics Nizhny Novgorod, a grant from the Government of the Russian Federation (Contract No. 075-15-2019-1931).

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  1. Vyacheslav Grines, Andrei Morozov and Olga Pochinka have contributed equally to this work.

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  1. International Laboratory DSA, NRU HSE, B. Pecherskaya Street 25/12, Nizhniy Novgorod, 603155, Nizhegorodskaya oblast, Russian Federation

    Vyacheslav Grines, Andrei Morozov & Olga Pochinka

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  1. Vyacheslav Grines
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Correspondence to Andrei Morozov.

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Grines, V., Morozov, A. & Pochinka, O. Determination of the Homotopy Type of a Morse-Smale Diffeomorphism on an Orientable Surface by a Heteroclinic Intersection. Qual. Theory Dyn. Syst. 22, 120 (2023). https://doi.org/10.1007/s12346-023-00809-9

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