On the Lefschetz zeta function for quasi-unipotent maps on the n-dimensional torus. II: The general case

We provide a general formula and give an explicit expression of the Lefschetz zeta function for any quasi-unipotent map on the n-dimensional torus. The Lefschetz zeta function is used to characterize the minimal set of Lefschetz periods for Morse–Smale diffeomorphisms on the n-dimensional torus; we completely describe this set, for different families containing infinitely many Morse–Smale diffeomorphisms. Moreover we show that for any given odd integer, there are Morse–Smale diffeomorphisms such that the corresponding minimal set of Lefschetz periods consists of all square free divisors of the given number. The results of the present article generalize the previous results of Berrizbeitia–Sirvent [7]. The methods used are based on the arithmetical properties of the number n.