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.
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).
Citation:
Timur V. Medvedev, Elena V. Nozdrinova, Olga V. Pochinka, “Components of Stable Isotopy Connectedness
of Morse – Smale Diffeomorphisms”, Regul. Chaotic Dyn., 27:1 (2022), 77–97
\Bibitem{MedNozPoc22}
\by Timur V. Medvedev, Elena V. Nozdrinova, Olga V. Pochinka
\paper Components of Stable Isotopy Connectedness
of Morse – Smale Diffeomorphisms
\jour Regul. Chaotic Dyn.
\yr 2022
\vol 27
\issue 1
\pages 77--97
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