Self-Similarity in the Collection of w-Limit Sets

Let w be the map which takes (x, f) in I x C(I x I) to the w-limit set w(x, f) with L the map taking f in C(I, I) to the family of w-limit sets {w(x, f) : x is an element of I}. We study R(w) = {w(x, f) : (x, f) is an element of I x C(I, I)}, the range of w, and R(L) = {L(f) : f is an element of C(I, I)}, the range of L. In particular, R(w) and its complement are both dense, R(w) is path-connected, and R(w) is the disjoint union of a dense G(delta) set and a first category F-sigma set. We see that R(L) is porous and path-connected, and its closure contains K = {F subset of [0, 1] : F is closed}. Moreover, each of the sets R(w) and R(L) demonstrates a self-similar structure.