We show that continuous triangular maps of the square I1, F: (x, y) → (f(x), g(x, y)), exhibit phenomena impossible in the one-dimensional case. In particular: (1) A triangular map F with zero topological entropy can have a minimal set containing an interval {a} × I, and can have recurrent points that are not uniformly recurrent; this solves two problems by S.F. Kolyada.

(2) In the class of mappings satisfying Per(F) = Fix(F), there are non-chaotic maps with positive sequence topological entropy and chaotic maps with zero sequence topological entropy.

We prove that an infinite W ⊂ (0, 1) is an ω-limit set for a continuous map ƒ of [0,1] with zero topological entropy iff W = Q ∪ P where Q is a Cantor set, and P is countable, disjoint from Q, dense in W if non-empty, and such that for any interval J contiguous to Q, card (J ∩ P) ≤ 1 if 0 or 1 is in J, and card (J ∩ P) ≤ 2 otherwise. Moreover, we prove a conjecture by A. N. Šarkovskii from 1967 that P can contain points from infinitely many orbits, and consequently, that the system of ω-limit sets containing Q and contained in W, can be uncountable.

We find a class of C∞ maps of an interval with zero topological entropy and chaotic in the sense of Li and Yorke.

Consider the continuous mappings f from a compact real interval to itself. We show that when f has a positive topological entropy (or equivalently, when f has a cycle of order ≠ 2n, n = 0, 1, 2, …) then f has a more complex behaviour than chaoticity in the sense of Li and Yorke: something like strong or uniform chaoticity, distinguishable on a certain level ɛ > 0. Recent results of the second author then imply that any continuous map has exactly one of the following properties: It is either strongly chaotic or every trajectory is approximable by cycles. Also some other conditions characterizing chaos are given.