A splitting result on transitivity for a class of n-dimensional maps

作者：Linero Bas Antonio; Soler Lopez Gabriel

来源：Nonlinear Dynamics, 2016, 84(1): 163-169.

DOI：10.1007/s11071-015-2311-y

• Cyclic permutation
• Cyclically permuted direct product map
• Transitivity
• Mixing
• Weakly mixing
• Total transitivity
• Splitting

We obtain some results on transitivity for cyclically permuted direct product maps, that is, maps of the form F (x(1), x(2),..., x(n)) = (f(sigma(1)) (x(sigma(1))), f(sigma(2)) (x(sigma(2))),..., f(sigma(n)) (x(sigma(n)))) defined from the Cartesian product I-n onto itself, where I = [0, 1], sigma is a cyclic permutation of {1, 2,..., n} (n >= 2) and each map f(sigma(j)) : I -> I is continuous, j is an element of {1,..., n}. In particular, we prove that for n >= 3 the transitivity of F is equivalent to the total transitivity, and if n = 2, we give a splitting result for transitive maps. Moreover, we extend well-known properties of transitivity from interval maps to cyclically permuted direct product maps. To do it, we use the strong link between F and the compositions phi(j) = f(sigma(j) o ..... o f(sigma)(n)(j), j is an element of {1,..., n}.

• 出版日期2016-4