A triangular map with homoclinic orbits and no infinite ω-limit set containing periodic points

Recently, Forti, Paganoni and Smital constructed an example of a triangular map of the unite square, F(x, y) = (f (x), g(x, y)), possessing periodic orbits of all periods and such that no infinite omega-limit set of F contains a periodic point. In this note we show that the above quoted map F has a homoclinic orbit. As a consequence, we answer in the negative the problem presented by A.N. Sharkovsky in the eighties whether, for a triangular map of the square, existence of a homoclinic orbit implies the existence of an infinite omega-limit set containing a periodic point. It is well known that, for a continuous map of the interval, the answer is positive. (C) 2005 Elsevier B.V. All rights reserved.