Mapping Borel Sets onto Balls and Self-similar Sets by Lipschitz and Nearly Lipschitz Maps

摘要

If X is an analytic metric space satisfying a very mild doubling condition, then for any finite Borel measure μ on X there is a set N ⊆ X such that μ(N) > 0, an ultrametric space Z and a Lipschitz bijection ϕ : N → Z whose inverse is nearly Lipschitz, i.e., β-Hölder for all β < 1. As an application it is shown that a Borel set in a Euclidean space maps onto [0, 1]^{n by a nearly Lipschitz map if and only if it cannot be covered by countably many sets of Hausdorff dimension strictly below n. The argument extends to analytic metric spaces satisfying the mild condition. Further generalization replaces cubes with self-similar sets, nearly Lipschitz maps with nearly Hölder maps and integer dimension with arbitrary finite dimension.}