A note on the dynamics of cyclically permuted direct product maps

We obtain some results on the dynamics of mapsF(x1,x2,…,xn)=(fσ(1)(xσ(1)),fσ(2)(xσ(2)),…,fσ(n)(xσ(n)))F(x1,x2,…,xn)=(fσ(1)(xσ(1)),fσ(2)(xσ(2)),…,fσ(n)(xσ(n))) (we call them cyclically permuted direct product maps), defined from the Cartesian product X1×X2×⋯×XnX1×X2×⋯×Xn into itself, where X1,X2,…,XnX1,X2,…,Xn are general topological spaces, each map fσ(i):Xσ(i)→Xifσ(i):Xσ(i)→Xi is continuous, i=1,…,ni=1,…,n, and σ   is a cyclic permutation of {1,2,…,n}{1,2,…,n}, n≥2n≥2. We study the topics of (totally) topological transitivity and (weakly) topological mixing for cyclically permuted direct product maps from the following point of view: we analyze the relationship between the dynamics of F   and that of the compositions fσ(i)∘…∘fσn(i)fσ(i)∘…∘fσn(i), i∈{1,…,n}i∈{1,…,n}.