On the Lefschetz Zeta Function for a Class of Toral Maps

Abstract

In the present article we study the periodic structure of a class of maps on the n -dimensional torus such that the eigenvalues of the induced map on the first homology are dilations of roots of unity. This family of maps shares some properties with the family of the quasi-unipotent maps. We compute their Lefschetz numbers, and show that the Lefschetz numbers of period m are non-zero, for all m ’s, in the case that n is an odd prime. Moreover we give an explicit expression for their corresponding Lefschetz zeta functions. We conjecture that in any dimension the Lefschetz numbers of period m are non-zero, for all m .