Periodic Structure of Transversal Maps on $\mathbb{C }$P$^{n}$, $\mathbb{H }$P$^{n}$ and $\mathbb{S }^{p}\times \mathbb{S }^{q}$

Abstract

A $\mathcal{C }^{1}$ map $f:\mathbb{M }\rightarrow \mathbb{M }$ is called transversal if for all $m\in \mathbb{N }$ the graph of $f^{m}$ intersects transversally the diagonal of $M\times M$ at each point $(x,x)$ being $x$ a fixed point of $f^m$. Let $\mathbb{C }$P$^{n}$ be the $n$-dimensional complex projective space, $\mathbb{H }$P$^{n}$ be the $n$-dimensional quaternion projective space and $\mathbb{S }^{p}\times \mathbb{S }^{q}$ be the product space of the $p$-dimensional with the $q$-dimensional spheres, $p\ne q$. Then for the cases $\mathbb{M }$ equal to $\mathbb{C }$P$^{n}$, $\mathbb{H }$P$^{n}$ and $\mathbb{S }^{p}\times \mathbb{S }^{q}$ we study the set of periods of $f$ by using the Lefschetz numbers for periodic points.