On a Classification of Periodic Maps on the 2-Torus

    Received 10 April 2022; accepted 10 June 2022; published 27 July 2022

    2023, Vol. 19, no. 1, pp.  91-110

    Author(s): Baranov D. A., Grines V. Z., Pochinka O. V., Chilina E. E.

    In this paper, following J. Nielsen, we introduce a complete characteristic of orientationpreserving periodic maps on the two-dimensional torus. All admissible complete characteristics were found and realized. In particular, each of the classes of orientation-preserving periodic homeomorphisms on the 2-torus that are nonhomotopic to the identity is realized by an algebraic automorphism. Moreover, it is shown that the number of such classes is finite. According to V. Z. Grines and A.Bezdenezhnykh, any gradient-like orientation-preserving diffeomorphism of an orientable surface is represented as a superposition of the time-1 map of a gradient-like flow and some periodic homeomorphism. Thus, the results of this work are directly related to the complete topological classification of gradient-like diffeomorphisms on surfaces.
    Keywords: gradient-like flows and diffeomorphisms on surfaces, periodic homeomorphisms, torus
    Citation: Baranov D. A., Grines V. Z., Pochinka O. V., Chilina E. E., On a Classification of Periodic Maps on the 2-Torus, Rus. J. Nonlin. Dyn., 2023, Vol. 19, no. 1, pp.  91-110

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