О структуре пространства орбит каскадов Морса-Смейла сферы

Динамические системы. 2018. Т. 2. № 15. С. 159–172.

Gurevich E., Смирнова А. С.

We consider a class $G$ of Morse-Smale diffeomorphisms  on the sphere $S^n$ of dimension $n\geq 4$ such that invariant manifolds of different saddle periodic points of any diffeomorphisms from $G$ have no intersection. Dynamics of an arbitrary  diffeomorphism $f\in G$ can be represented as ``sink-source'' dynamics where the  ``sink'' $A_f$ (the ``source'' $R_f$) is  the connected unions of one- and zero-dimensional   unstable (stable) manifolds.  We  study a structure of  the space  $V_f=S^n\setminus (A_f\cup R_f)$ and  the topology of embedding in $V_f$ of separatrices of dimension $(n-1)$.  We prove that the orbit space $\widehat{V}_f=V_f/_f$ is homeomorphic to the direct product $\mathbb{S}^{n-1}\times \mathbb{S}^1$, and the projection $l_\sigma\subset \widehat{V}_f$ of $(n-1)$-dimensional separatrix of a  saddle periodic point $\sigma$ is   either homeomorphic  to the direct product $\mathbb{S}^{n-2}\times \mathbb{S}^1$ and bounds in $\widehat{V}_f$  a manifold homeomorphic to  $\mathbb{B}^{n-1}\times \mathbb{S}^1$, or homeomorphic to a non-oriented locally-trivial fiber bundle under the circle $\mathbb{S}^1$ with the fiber $\mathbb{S}^{n-2}$, and  such the manifold may by only one amount all projection of the separatrices.

Keywords: топологическая сопряженность, топологическая классификация, topological conjugacy, topological classification, Morse-Smale cascades, Каскады Морса-Смейла

Publication based on the results of: Topology and chaos in the dynamics of systems, foliations, and deformations of Lie algebras (2018)