Triangular maps non-decreasing on the fibers.

Abstract

There is a list of about 50 properties which characterize continuous maps of the interval with zero topological entropy. Most of them were proved by A.N. Sharkovsky [cf., e.g., Sharkovsky et al., Dynamics of One-Dimensional Mappings, Kluwer 1997]. It is also well known that only a few of these properties remain equivalent for continuous maps of the square. Recall, e.g., the famous Kolyada's example of a triangular map of type $2^\infty$ with positive topological entropy. In 1989 Sharkovsky formulated the problem to classify these conditions in a special case of triangular maps of the square. The present paper is a step toward the solution. In particular, we give a classification of 23 conditions in the case of triangular maps of the square which are non-decreasing on the fibers. We show that the weakest is"no homoclinic trajectory", the two strongest, mutually incomparable, are "map restricted to the set of chain recurrent points is not Li and Yorke chaotic" and "every $\omega$-limit set contains a unique minimal set"