Components of Stable Isotopy Connectedness of Morse – Smale Diffeomorphisms

Regular and Chaotic Dynamics. 2022. Vol. 27. No. 1. P. 77–97.

Medvedev T. V., Nozdrinova E., Pochinka O.

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: Morse – Smale diffeomorphism, stable arc

Publication based on the results of: Теория динамических систем и ее приложения (2023)