Abstract

We compute the Lefschetz zeta function for quasi-unipotent maps on the n-dimensional torus, using arithmetical properties of the number n. In particular we compute the Lefschetz zeta function for quasi-unipotent maps, such that the characteristic polynomial of the induced map on the first homology group is the (n + 1)th cyclotomic polynomial, when n + 1 is an odd prime. These computations involve fine combinatorial properties of roots of unity. We also show that the Lefchetz zeta functions for quasi-unipotent maps on T-n are rational functions of total degree zero. We use these results in order to characterized the minimal set of Lefschetz periods for C-1 quasi-unipotent maps on T-n, having finitely many periodic points all of them hyperbolic. Among this class of maps are the Morse-Smale diffeomorphisms of T-n.