Many-Dimensional Morse–Smale Diffeomeophisms with a Dominant Saddle

The article defines a class of Morse–Smale diffeomorphisms with a dominant saddle and provides necessary and sufficient conjugacy conditions for such diffeomorphisms. We show that the polar Morse–Smale diffeomorphisms of the $n$-dimensional sphere $\mathbb S^n$, $n\ge4$, whose nonwandering set consists of four points have dominant saddles. As a corollary, we obtain necessary and sufficient conjugacy conditions for such diffeomorphisms. We give examples of polar Morse–Smale diffeomorphisms $\mathbb S^n\to\mathbb S^n$ with such a nonwandering set.