Coupled-Expanding Maps for Irreducible Transition Matrices

In this paper, strictly A-coupled-expanding maps in bounded and closed subsets of complete metric spaces are investigated, where A = (aij) is an m × m irreducible transition matrix with one row-sum no less than 2. A map f is said to be strictly A-coupled-expanding in m sets Vi if f(Vi) ⊃ Vj whenever aij = 1 and the distance between any two different sets of these Vi is positive. A new result on the subshift for matrix A is obtained. Based on this result, two criteria of chaos are established, which generalize and relax the conditions of some existing results. These maps are proved to be chaotic either in the sense of both Li–Yorke and Wiggins or in the sense of both Li–Yorke and Devaney. One example is provided to illustrate the theoretical results with a computer simulation for demonstration.