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Components of Stable Isotopy Connectedness of Morse – Smale Diffeomorphisms

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Abstract

In 1976 S. Newhouse, J. Palis and F. Takens introduced a stable arc joining two structurally stable systems on a manifold. Later in 1983 they proved that all points of a regular stable arc are structurally stable diffeomorphisms except for a finite number of bifurcation diffeomorphisms which have no cycles, no heteroclinic tangencies and which have a unique nonhyperbolic periodic orbit, this orbit being the orbit of a noncritical saddle-node or a flip which unfolds generically on the arc. There are examples of Morse – Smale diffeomorphisms on manifolds of any dimension which cannot be joined by a stable arc. There naturally arises the problem of finding an invariant defining the equivalence classes of Morse – Smale diffeomorphisms with respect to connectedness by a stable arc. In the present review we present the classification results for Morse – Smale diffeomorphisms with respect to stable isotopic connectedness and obstructions to existence of stable arcs including the authors’ recent results.

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Notes

  1. Indeed, S. Matsumoto’s proved Theorem 4 for the so-called simple arcs which include the stable arc as a special case.

  2. A smooth 3-manifold is simple if it is either irreducible (every smooth 2-sphere bounds the 3-ball in it) or it is homeomorphic to \(S^{2}\times S^{1}\).

  3. A \(C^{0}\)-map \(g:B\to X\) is called a topological embedding of the topological manifold \(B\) to the manifold \(X\) if it homeomorphically sends \(B\) to the subspace \(g(B)\) with the topology induced from \(X\). The image \(A=g(B)\) is called the topologically embedded manifold. Notice that a topologically embedded manifold is not a topological submanifold in general. If \(A\) is a submanifold, then it is said to be tame or tamely embedded. Otherwise \(A\) is called wild or wildly embedded; the points in which the conditions of the topological manifold are not satisfied are called the points of wildness.

  4. Here the sphere \(S^{3}\) is identified with a specific unitary group \(SU(2)=\left\{\begin{pmatrix}\alpha&-\bar{\beta}\\ \beta&\bar{\alpha}\end{pmatrix}:\alpha,\beta\in\mathbb{C},|\alpha|^{2}+|\beta|^{2}=1\right\}\) where the dash means complex conjugate.

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Funding

The research on the obstructions to existence of a stable arc between isotopic Morse – Smale diffeomorphisms is supported by RSF (Grant No. 21-11-00010), and the research on components of the stable connection of gradient-like diffeomorphisms of surfaces is supported by the Laboratory of Dynamical Systems and Applications NRU HSE, by the Ministry of Science and Higher Education of the Russian Federation (ag. 075-15-2019-1931) and by the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS” (project 19-7-1-15-1).

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Authors and Affiliations

  1. Laboratory of Algorithms and Technologies for Network Analysis, HSE University, ul. Rodionova 136, 603093, Nizhny Novgorod, Russia

    Timur V. Medvedev

  2. International Laboratory of Dynamical Systems and Applications, HSE University, ul. Bolshaya Pecherckaya 25/12, 603155, Nizhny Novgorod, Russia

    Elena V. Nozdrinova & Olga V. Pochinka

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  1. Timur V. Medvedev
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  2. Elena V. Nozdrinova
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  3. Olga V. Pochinka
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Correspondence to Timur V. Medvedev, Elena V. Nozdrinova or Olga V. Pochinka.

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37C15, 37D15

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Medvedev, T.V., Nozdrinova, E.V. & Pochinka, O.V. Components of Stable Isotopy Connectedness of Morse – Smale Diffeomorphisms. Regul. Chaot. Dyn. 27, 77–97 (2022). https://doi.org/10.1134/S1560354722010087

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