Stable Arcs Connecting Polar Cascades on a Torus

Russian Journal of Nonlinear Dynamics. 2021. Vol. 17. No. 1. P. 23–37.

Pochinka O., Nozdrinova E.

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.

Keywords: gradient-like diffeomorphism, stable arc, устойчивая дуга, saddle-node, седло-узел, two-dimensional torus, градиенто-подобный диффеоморфизм, двумерный тор

Publication based on the results of: Теория динамических систем и ее приложения (2021)