Bau-Sen Du

Copyright © 2004 Bau-Sen Du. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let n≥2 be an integer and let P={1,2,…,n,n+1}. Let Zp denote the finite field {0,1,2,…,p−1}, where p≥2 is a prime. Then every map σ on P determines a real n×n Petrie matrix Aσ which is known to contain information on the dynamical properties such as topological entropy and the Artin-Mazur zeta function of the linearization of σ. In this paper, we show that if σ is a cyclic permutation on P, then all such matrices Aσ are similar to one another over Z2 (but not over Zp for any prime p≥3) and their characteristic polynomials over Z2 are all equal to ∑k=0nxk. As a consequence, we obtain that if σ is a cyclic permutation on P, then the coefficients of the characteristic polynomial of Aσ are all odd integers and hence nonzero.