DOI 10.15507/2079-6900.23.202102.171–184

Original article

ISSN 2587-7496 (Print)

ISSN 2079-6900 (Online)

MSC2020 37E30

Realization of homotopy classes of torus homeomorphisms by the simplest structurally stable diffeomorphisms

A. I. Morozov

National Research University «Higher School of Economics» (Nizhny Novgorod, Russian Federation)

Abstract. According to Thurston’s classification, the set of homotopy classes of orientation-preserving homeomorphisms of orientable surfaces is split into four disjoint subsets. A homotopy class from each subset is characterized by the existence of a homeomorphism called Thurston’s canonical form, namely: a periodic homeomorphism, a reducible nonperiodic homeomorphism of algebraically finite order, a reducible homeomorphism that is not a homeomorphism of an algebraically finite order, and a pseudo-Anosov homeomorphism. Thurston’s canonical forms are not structurally stable diffeomorphisms. Therefore, the problem naturally arises of constructing the simplest (in a certain sense) structurally stable diffeomorphisms in each homotopy class. In this paper, the problem posed is solved for torus homeomorphisms. In each homotopy class, structurally stable representatives are analytically constructed, namely, a gradient-like diffeomorphism, a Morse-Smale diffeomorphism with an orientable heteroclinic, and an Anosov diffeomorphism, which is a particular case of a pseudo-Anosov diffeomorphism.

Key Words: Nielsen-Thurston theory, homotopic classes of mappings, realization of diffeomorphisms, algebraic mappings

For citation: A. I. Morozov. Realization of homotopy classes of torus homeomorphisms by the simplest structurally stable diffeomorphisms. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 23:2(2021), 171–184. DOI: https://doi.org/10.15507/2079-6900.23.202102.171–184

Submitted: 21.03.2021; Revised: 23.04.2021; Accepted: 26.05.2021

Information about the author:

Andrei I. Morozov, Research Assistant, International Laboratory of Dynamical Systems and Applications, National Research University «Higher School of Economics» (25/12 Bolshaya Pecherskaya St., Nizhny Novgorod 603155, Russia), ORCID: https://orcid.org/0000-0003-3125-1825, andreifrostnn@gmail.com

The author have read and approved the final manuscript.

Conflict of interest: The author declare no conflict of interest.

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