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Timur Medvedev


Medvedev T. V., Nozdrinova E. V., Pochinka O. V.
Components of Stable Isotopy Connectedness of Morse – Smale Diffeomorphisms
2022, vol. 27, no. 1, pp.  77-97
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.
Keywords: stable arc, Morse – Smale diffeomorphism
Citation: Medvedev T. V., Nozdrinova E. V., Pochinka O. V.,  Components of Stable Isotopy Connectedness of Morse – Smale Diffeomorphisms, Regular and Chaotic Dynamics, 2022, vol. 27, no. 1, pp. 77-97

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