Periods of self-maps on \({\mathbb{S}}^2\) via their homology (Q6585753)

Let \(\mathbb{S}^2\) be a 2-dimensional sphere and \(f\) be a continuous self-map on \(\mathbb{S}^2\). This paper summarizes results on the periodic orbits of different kinds of self-maps obtained by the author and his collaborators in the last two decades. In the past, they investigated \(C^1\) maps such that all their periodic orbits are hyperbolic, transversal maps, holomorphic maps and transversal holomorphic maps by using the Lefschetz numbers associated with the self-maps. The author highlights the observation that when a self-map increases its structure, the number of periodic orbits provided by its action on the homology increases too.