Strange triangular maps of the square

Abstract

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.