Strange distributionally chaotic triangular maps III

In the class T of triangular maps of the square we consider the strongest notion of distributional chaos, DC1, originally introduced by Schweizer and Smítal [Trans Amer Math Soc 1994;344:737-854] for continuous maps of the interval. We show that a map F∈T is DC1 if F has a periodic orbit with period ≠ 2n, for any n ⩾ 0. Consequently, a map in T is DC1 if it has a homoclinic trajectory. This result is important since in general systems like T, positive topological entropy itself does not imply DC1. It contributes to the solution of a long-standing open problem of A. N. Sharkovsky concerning classification of triangular maps of the square.