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

TL;DR

This paper analyzes the periodic behavior of certain smooth self-maps on products of spheres and Lie groups, using Lefschetz fixed point theory to characterize minimal periods and conditions for infinite periodic points.

Contribution

It provides a complete characterization of Lefschetz periods for Morse-Smale diffeomorphisms and conditions for maps to have infinitely many periodic points on these spaces.

Findings

Complete characterization of minimal Lefschetz periods for Morse-Smale diffeomorphisms.

Conditions for maps to have infinitely many periodic points.

Description of the period set for transversal maps.

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.