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.