Conformal Dimension of Measures and Quasisymmetric Dimension Reduction

Abstract

We prove that the conformal dimension of every locally finite Borel measure is either zero or infinite. The main ingredient is a quasisymmetric dimension-reduction theorem: every full-support probability measure of finite Hausdorff dimension on a separable metric space admits quasisymmetrically equivalent metrics in which its Hausdorff dimension is arbitrarily small. In particular, every locally finite Borel measure on a doubling metric space has conformal dimension zero. We also prove that for every $n\geq 1$ and $p>0$ there are a metric space $X$, a quasisymmetric homeomorphism $f:[0,1]^n\to X$, and a Borel set $E\subset[0,1]^n$ such that $\dim_H f(E)\leq p$ and $\dim_H([0,1]^n\setminus E)\leq n-1+p$. The second bound is sharp up to $p$: if $\dim_H f(E)<1$, then $\dim_H([0,1]^n\setminus E)\geq n-1$.