On the class of square Petrie matrices induced by cyclic permutations

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.