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 Z p 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 A σ 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 A σ are similar to one another over Z 2 (but not over Z p for any prime p ≥ 3 ) and their characteristic polynomials over Z 2 are all equal to ∑ k = 0 n x k . As a consequence, we obtain that if σ is a cyclic permutation on P , then the coefficients of the characteristic polynomial of A σ are all odd integers and hence nonzero.