Self-Similarity in the Collection of ω-Limit Sets

摘要

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