Permutation Groups and Periodicity of Systems of Difference Equations

Abstract

Let k ∈ N, Z k = {1, 2,. .. , k} and S k be the group of all permutations on Z k. Let π ∈ S k be of order l and fi be a function from a nonempty set X into itself, i = 1,. .. , k. In this paper, we show that a sufficient condition for a system of difference equations x (1) n+1 = f1(x (π(1)) n−s), x (2) n+1 = f2(x (π(2)) n−s),. .. x (k) n+1 = f k (x (π(k)) n−s), n ∈ Z ≥0 , to be periodic with a period d is that each difference equation y n+l(s+1)−s = gi(yn−s), n ∈ Z ≥0 , is periodic, i = 1,. .. , k, with a period di that divides d. Here, gi is defined by gi = fif π(i). .. f π l−1 (i) , i = 1,. .. , k. Finally, the periodicity of many systems of rational difference equations is established.