Cantor Sets, Binary Trees, and Lipschitz Circle Homeomorphisms

摘要

Let A1 and A2 be disjoint compact subintervals of [0, 1], and let L be the small- est compact interval containing A1 U A2 . Let S : A1i U A2 → L be a mapping such that the restrictions S|a1 are affine surjections onto L for each i. Then we define the affine Cantor set Ks by Ks ≡ { x ∈ L : S1 (x) is contained in A1 U A1 for all ≥ 1 } . We call this a two-branched alffine Cantor set. If we replace the restriction that S be locally affine with the requirement that |S'|'> l, than Ks is called a two- branched hyperkolic Cantor set. (This is sometimes also called a dynamically de- fined Cantor set, a selfsimilar Cantor set, or a ``cookie cutter''.) A k-branched affine Cantor set or hyperbolic Cantor set is defined similarly for any k ≥ 2.