Distributional chaos for triangular maps

Abstract

In this paper we exhibit a triangular map F of the square with the following properties: (i) F is of type 2 ∞ but has positive topological entropy; we recall that similar example was given by Kolyada in 1992, but our argument is much simpler. (ii) F is distributionally chaotic in the wider sense, but not distributionally chaotic in the sense introduced by Schweizer and Smítal [Trans. Amer. Math. Soc. 344 (1994) 737]. In other words, there are lower and upper distribution functions Φ xy and Φ xy ∗ generated by F such that Φ xy ∗≡1 and Φ xy (0 +)<1, and no distribution functions Φ uv , and Φ uv ∗ such that Φ uv ∗≡1 and Φ uv ( t)=0 whenever 0< t< ϵ, for some ϵ>0. We also show that the two notions of distributional chaos used in the paper, for continuous maps of a compact metric space, are invariants of topological conjugacy.