Abstract

In this talk we mention the importance of the Lefschetz zeta function in the study of the periodic structure of a map. We give an explicit and closed formula for quasi-unipotent maps on the $n$-dimensional torus. A map is called quasi-unipotent if all eigenvalues of the induced maps on homology are roots of unity. Later we use this formula to obtain information on the periodic structure of a large class of differentiable maps, among them Morse-Smale diffeomorphisms, on the torus. The method used in the computation of the Lefschetz zeta function is based on arithmetical properties of the number $n$. We will discuss extensions of techniques used in the study of maps on other spaces.