Let 𝒦 be the class of
compact subsets of I = [0,1], furnished with the Hausdorf metric. Let f ∈ C(I,I).
We study the map ωf: I →𝒦 defined as ωf(x) = ω(x,f), the ω-limit set
of x under f. This map is rarely continuous, and is always in the second
Baire class. Those f for which ωf 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 ω-limit
sets with isolated points. We also discuss methods of constructing functions
with zero topological entropy exhibiting infinite ω-limit sets with various
properties.
