Existence of quasiconformal maps with maximal stretching on any given countable set.
2021
Quasiconformal maps are homeomorphisms with useful local distortion inequalities; infinitesimally, they map balls to ellipsoids with bounded eccentricity. This leads to a number of useful regularity properties, including quantitative Holder continuity estimates; on the other hand, one can use the radial stretches to characterize the extremizers for Holder continuity. In this work, given any bounded countable set in $\mathbb{R}^d$, we will construct an example of a $K$-quasiconformal map which exhibits the maximum stretching at each point of the set. This will provide an example of a quasiconformal map that exhibits the worst-case regularity on a surprisingly large set, and generalizes constructions from the planar setting into $\mathbb{R}^d$.
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