Self-similarity in the collection of [omega]-limit sets

Let [omega] be the map which takes (x, f) in I x C(I x I) to the [omega]-limit set [omega](x, f) with L the map taking f in C(I, I) to the family of [omega]-limit sets {[omega](x, f) : x [member of] I}. We study R([omega]) = {[omega](x, f) : (x, f) [member of] I x C(I, I)}, the range of [omega], and R(L) = {L(f) : f [member of] C(I, I)}, the range of L. In particular, R([omega]) and its complement are both dense, R([omega]) is path-connected, and R([omega]) is the disjoint union of a dense [G.sub.[delta]] set and a first category [F.sub.[sigma]] set. We see that R(L) is porous and path-connected, and its closure contains K = {F [subset or equal to] [0, 1] : F is closed}. Moreover, each of the sets R([omega]) and R(L) demonstrates a self-similar structure. Keywords. Continuous self-map, [omega]-limit set, porous set. Mathematics Subject Classification (2010). Primary 26A18, 37B99, secondary 54H20