COUPLED-EXPANDING MAPS AND MATRIX SHIFTS

Abstract

For an irreducible transition matrix A of size m × m, which is not a permutation, a map f : X → X is said to be strictly A-coupled-expanding if there are nonempty sets V1,…, Vm ⊂ X such that the distance between any two of them is positive and f(Vi) ⊃ Vj holds whenever aij = 1. This paper presents two theorems that give sufficient conditions for a strictly A-coupled-expanding map to be chaotic on part of its domain in the sense of, respectively, Auslander and Yorke and Devaney. These results improve on the work of Zhang and Shi [2010]. An example is provided to illustrate that the class of maps the new theorems apply to is significantly wider.