Non - Removable Sets for Quasiconformal and Locally Bilipschitz Mappings in R

We give an example of a totally disconnected set E ⊂ R which is not removable for quasiconformal homeomorphisms, i.e., there is a homeomorphism f of R to itself which is quasiconformal off E, but not quasiconformal on all of R. The set E may be taken with Hausdorff dimension 2. The construction also gives a non-removable set for locally biLipschitz homeomorphisms. 1. Statement of results If a homeomorphism of R to itself is quasiconformal except on a compact set E, does it have to be quasiconformal on all of R? If so, E is called removable for quasiconformal mappings. The purpose of this paper is to construct examples of non-removable sets in R 3 which are as small as possible, both topologically (they are totally disconnected) and metrically (they have Hausdorff dimension 2). A mapping is called quasiconformal on Ω ⊂ R if there is an M < ∞ so that lim sup r→0 sup|x−y|=r |f(x)− f(y)| inf |x−y|=r |f(x)− f(y)| ≤ M ∀x ∈ Ω. (See [12] or Theorem 34.1 of [22].) Our method will actually give non-removable sets for an even more restrictive class of mappings. We say that a mapping is locally biLipschitz on Ω if there is an M < ∞ so that for every x ∈ Ω there is an r = r(x) > 0 so that |x − y| < r implies M ≤ |f(x)− f(y)| |x− y| ≤ M. Such mappings are also called bounded length distortion (e.g., [23], [24]) or local quasiisometries (e.g., [9], [15]). If a quasiconformal mapping is biLipschitz on dense open set then it is globally biLipschitz, and hence a non-removable set for the biLipschtiz maps is also non-removable for quasiconformal maps. Theorem 1.1. There is a totally disconnected set E ⊂ R which is nonremovable for locally biLipschitz (and hence for quasiconformal) maps. If φ(t) = o(t) then we may choose E and f so that H(E) = H(f(E)) = 0. Here H denotes the φ-Hausdorff measure, i.e., H(E) = lim δ→0 [inf{ ∑ j φ(rj), E ⊂ ∪jB(xj , rj), rj ≤ δ}]. 1991 Mathematics Subject Classification. 30C65.