Abstract

ondider a continuous maps f from a compact interval into itself. Two points x,y form a Li-Yorke pair if the limsup of d(f^n(x), f^n(y)) is positive and the liminf of the same quantity is zero when n tends to infinity. If f has a horseshoe, and more generally if the topological entropy of f is positive, then there exist (uncountably many) Li-Yorke pairs. The converse is not true and there exist Li-Yorke chaotic maps of zero topological entropy. The existence of Li-Yorke pairs is a week property. However, when the set of Li-Yorke pairs is dense, then we show that the map f has a precise structure, and in particular f2 has a horseshoe. We will illustrate these results with examples.