On the Lefschetz zeta function and the minimal sets of Lefschetz periods for Morse–Smale diffeomorphisms on products of ℓ-spheres

Abstract

We provide a general formula and give an explicit expression of the Lefschetz zeta function for any quasi-unipotent map on the space X=Sℓ×⋯×︸n−timesSℓ, with ℓ>1. Among the quasi-unipotent maps are Morse–Smale diffeomorphisms. The Lefschetz zeta function is used to characterize the minimal set of Lefschetz periods for Morse–Smale diffeomorphisms on X; we completely describe this set, for families containing infinitely many Morse–Smale diffeomorphisms. The results of the present article are based on the techniques used in [5], in the computation of the Lefschetz zeta function for quasi-unipotent self maps on the n-dimensional torus.