On the classes of stable isotopic connectivity of polar surface cascades

Today, one of the significant problems in the theory of dynamical systems is the problem of the existence of an arc with no more than a countable (finite) number of bifurcations connecting structurally stable systems (Morse-Smale systems) on manifolds. This problem is included in the list of fifty Palis-Pugh problems in their work under the number 33. In 1976, S. Newhouse, J. Palis, F. Tackens 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 little movement. 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. It follows from the result of the work of J. Fleitas that a simple arc constructed by Newhouse and Peixoto can always be replaced by a stable one. For Morse – Smale diffeomorphisms given on manifolds of any dimension, examples of systems that cannot be connected by a stable arc are known. In this connection, the question naturally arises of finding an invariant that uniquely determines the equivalence class of the Morse – Smale diffeomorphism with respect to the connection relation by a stable arc (the component of stable connection). The main result of the paper is the proof of the following result: any diffeomorphism from the class in question is connected with a diffeomorphism, which is the product of two source-sink diffeomorphisms on a circle, a stable arc with a finite number of saddle-node bifurcations, where the class in question is a class of isotopic identically polar diffeomorphisms on a two-dimensional torus , under the assumption that all nonwandering points are motionless and have a positive orientation type.