Periods of Morse--Smale diffeomorphisms on ${\mathbb{S}}^n$, ${\mathbb{S}}^m\times {\mathbb{S}}^n$, $\mathbb{C}P^n}$ and $\mathbb{H}P^n}$

Abstract

We study the set of periods of the Morse--Smale diffeomorphisms on the $n$-dimensional sphere ${\mathbb{S}}^n$, on products of two spheres of arbitrary dimension ${\mathbb{S}}^m\times {\mathbb{S}}^n$ with $m\ne n$, on the $n$-dimensional complex projective space $\mathbb{C}\text{\textbf{P}}^n}$ and on the $n$-dimensional quaternion projective space $\mathbb{H}\text{\textbf{P}}^n}$. We classify the minimal sets of Lefschetz periods for such Morse--Smale diffeomorphisms. This characterization is done using the induced maps on the homology. The main tool used is the Lefschetz zeta function.

12 pages