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 Rus. J. Nonlin. Dyn., 2021, Volume 17, Number 1, Pages 23–37 (Mi nd739)

Nonlinear physics and mechanics

Stable Arcs Connecting Polar Cascades on a Torus

O. V. Pochinka, E. V. Nozdrinova

Higher School of Economics — Nizhny Novgorod, ul. B. Pecherskaya 25/12, Nizhny Novgorod, 603150 Russia

Abstract: The problem of the existence of an arc with at most countable (finite) number of bifurcations connecting structurally stable systems (Morse – Smale systems) on manifolds was included in the list of fifty Palis – Pugh problems at number 33.
In 1976 S. Newhouse, J.Palis, F.Takens introduced the concept of a stable arc connecting two structurally stable systems on a manifold. Such an arc does not change its quality properties with small changes. In the same year, S.Newhouse and M.Peixoto proved the existence of a simple arc (containing only elementary bifurcations) between any two Morse – Smale flows. From the result of the work of J. Fliteas it follows that the simple arc constructed by Newhouse and Peixoto can always be replaced by a stable one. For Morse – Smale diffeomorphisms defined on manifolds of any dimension, there are examples of systems that cannot be connected by a stable arc. In this connection, the question naturally arises of finding an invariant that uniquely determines the equivalence class of a Morse – Smale diffeomorphism with respect to the relation of connection by a stable arc (a component of a stable isotopic connection).
In the article, the components of the stable isotopic connection of polar gradient-like diffeomorphisms on a two-dimensional torus are found under the assumption that all non-wandering points are fixed and have a positive orientation type.

 Funding Agency Grant Number Russian Science Foundation 17-11-01041 Foundation for the Development of Theoretical Physics and Mathematics BASIS 19-7-1-15-1 This work is supported by the Russian Science Foundation under grant 17-11-01041, except of study of the dynamics of diffeomorphisms of the class under consideration supported by Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS” (project 19-7-1-15-1).

DOI: https://doi.org/10.20537/nd210103

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MSC: 37D15
Accepted:21.03.2021
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Citation: O. V. Pochinka, E. V. Nozdrinova, “Stable Arcs Connecting Polar Cascades on a Torus”, Rus. J. Nonlin. Dyn., 17:1 (2021), 23–37

Citation in format AMSBIB
\Bibitem{PocNoz21} \by O. V. Pochinka, E. V. Nozdrinova \paper Stable Arcs Connecting Polar Cascades on a Torus \jour Rus. J. Nonlin. Dyn. \yr 2021 \vol 17 \issue 1 \pages 23--37 \mathnet{http://mi.mathnet.ru/nd739} \crossref{https://doi.org/10.20537/nd210103} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=4240815} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85105225967}