On topological classification of Morse–Smale cascades by veans of combinatorial invariants

A diffeomorphism $f:M^n\rightarrow M^n$ of smooth closed manifold $M^n$ is called a Morse–Smale if its non-wandering set $\Omega_f$ is finite, consists of hyperbolic points, and for any points $p,q\in\Omega_f$ an intersection of the stable manifold of $p$ with the unstable manifold of $q$ is transversal (see for instance [1]).
Thanks to the finiteness of the set of non-wandering orbits it is possible to obtain topological classification in wide classes of Morse-Smale systems by means of combinatorial invariants describing the mutual arrangement of invariant manifolds. In first time this approach was applied by E.Leontovich and A.Mayer for clasification of flows with finite nimber of singular trajecoties on the two-dimensional sphere. Further this idea was developed by M.Peixoto, A.Oshemkov, V.Sharko, Y.Umanskii who solved similar problem for Morse–Smale flows on manifolds of dimension 2, 3 and greater, and by Ch.Bonatti, A.Bezdenezhnyich, V.Grines, V.Medvedev, R.Langevin, O.Pochinka, E.Gurevich for Morse-Smale cascades (see review [2] for references).
In the report we establish that Morse–Smale cascades without heteroclinical intersections defined on the sphere $\mathbb S^n$, $n\geqslant4$, also admit the complete topological classification in a combinatorial language. This result contrasts with a case of Morse–Smale cascades on three-dimensional manifolds (см. [2,3]).
Research was supported by Russian Science Foundation (project 17-11-01041).
References
[1] V.Grines, T.Medvedev, O.Pochinka, Dynamical systems on 2- and 3-manifolds. Switzerland. Springer International Publishing, 2016.
[2] V. Grines, E. Gurevich, E. Zhuzhoma, O. Pochinka, Classification of Morse–Smale systems and the topological structure of ambient manifolds, UMN, 74:1 (2019), 41–116.
[3] V.Grines, E.Gurevich, O.Pochinka, Combinatorial invariant for Morse–Smale cascades without heteroclinic intersections on the sphere $\mathbb S^n$, $n\geqslant4$, Math. Notes, 105:1 (2019), 136–141.