On the Lefschetz zeta function for quasi-unipotent maps on the n-dimensional torus

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 th cyclotomic polynomial, when 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 are rational functions of total degree zero. We use these results in order to characterized the minimal set of Lefschetz periods for quasi-unipotent maps on , having finitely many periodic points all of them hyperbolic. Among this class of maps are the Morse–Smale diffeomorphisms of .

Keywords::

• cyclotomic polynomial
• Lefschetz zeta function
• periodic point
• minimal set of Lefschetz periods
• Morse–Smale diffeomorphism
• quasi-unipotent map

Keywords::

• 37E15
• 37C25
• 11T22
• 37D15

Notes

^1. Email: [email protected]