On the periodic structure of $C^1$ self-maps on the product of spheres of different dimensions

Abstract

In the present article we study the periodic structure of some well-known classes of $C^1$ self-maps on the product of spheres of different dimensions: transversal maps, Morse-Smale diffeomorphisms and maps with all its periodic points hyperbolic. Our approach is via the Lefschetz fixed point theory. We give a complete characterization of the minimal set of Lefschetz periods for Morse-Smale diffeomorphisms on these spaces. We also consider $C^1$ maps with all its periodic points hyperbolic and we give conditions for these maps to have infinitely many periodic points. We describe the period set of the transversal maps on these spaces. Finally we applied these results to describe the periodic structure of similar classes of maps on compact, connected, simple connected and simple Lie groups.