Distributional chaos for $\mathbb{Z}^d$-actions

Abstract

In this paper, we study the relationship between the notions of distributional chaos and specification property defined for multidimensional time discrete dynamical systems. Essentially, we prove that a $\mathbb{Z}^d$-action on a compact metric space with weak specification property and with a pair of distal points admits a dense distributionally scrambled set of type 1.