Many-Dimensional Morse–Smale Diffeomeophisms with a Dominant Saddle

Mathematical notes. 2022. Vol. 111. No. 5-6. P. 870–878.

E. V. Zhuzhoma, V. S. Medvedev

We introduce high-dimensional Morse-Smale systems with king-saddles. Every saddle of such system has a separatrix with heteroclinic intersections. We get the necessary and sufficient conditions of conjugacy of Morse-Smale diffeomorphisms with king-saddles. For every polar Morse-Smale system with two saddles on the $n$-sphere $\mathbb{S}^n$, one proves the existence of a king-saddle. This allows to get the necessary and sufficient conditions of conjugacy for such Morse-Smale diffeomorphisms on $\mathbb{S}^n$, $n\geq 4$. We give a simple sufficient condition of the existence of heteroclinic intersections for Morse-Smale systems on $\mathbb{S}^3$.

Keywords: topological conjugacy, topological classification, Morse-Smale systems

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