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 $n\ge 1$ 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 ${\dim }_{H}f(E)\le p$ and ${\dim }_H([0,1{]}^n\setminus E)\le n-1+p$. The second bound is sharp up to $p$: if ${\dim }_{H}f(E)<1$, then ${\dim }_H([0,1{]}^n\setminus E)\ge n-1$.

29 pages