arXiv 1802.080952018-02-22math.CA

Mapping Borel sets onto balls and self-similar sets by Lipschitz and nearly Lipschitz maps

Ondřej Zindulka

英文摘要：

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 $N\subseteq X$ such that $μ(N)>0$, an ultrametric space $Z$ and a Lipschitz bijection $ϕ:N\to Z$ 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.

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