CHAOS IN TERMS OF THE MAP CHI-]OMEGA(CHI, F)

Let K be the class of compact subsets of I = [0, 1] , furnished with the Hausdorf metric. Let f is-an-element-of C(I, I) . We study the map omega(f) : I --> K defined as omega(f)(x) = omega(x ,f), the omega-limit set of x under f . This map is rarely continuous, and is always in the second Baire class. Those f for which omega(f) is in the first Baire class exhibit a form of nonchaos that allows scrambled sets but not positive entropy. This class of functions can be characterized as those which have no infinite omega-limit sets with isolated points. We also discuss methods of constructing functions with zero topological entropy exhibiting infinite omega-limit sets with various properties. 4