On the class of square Petrie matrices induced by cyclic permutations

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 ∑n k=0x. 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.