Porosity of the branch set of discrete open mappings with controlled linear dilatation

Abstract

Assume that $X$ and $Y$ are locally compact and locally doubling metric spaces, which are also generalized $n$-manifolds, that $X$ is locally linearly locally $n$-connected, and that $Y$ has bounded turning. In this paper, addressing Heinonen's ICM 02 talk, we study the geometry of the branch set $\mathcal{B}_f$ of a quasiregular mapping between metric $n$-manifolds. In particular, we show that $\mathcal{B}_f\cap \{x\in X:H_f(x)<\infty\}$ is countably porous, as is its image $f\big(\mathcal{B}_f\cap \{x\in X:H_f(x)<\infty\}\big)$. As a corollary, $\mathcal{B}_f\cap \{x\in X:H_f(x)<\infty\}$ and its image are null sets with respect to any locally doubling measures on $X$ and Y, respectively. Moreover, if either $H_f(x)\leq H$ or $H_f^*(x)\leq H^*$ for all $x\in X$, then both $\mathcal{B}_f$ and $f\big(\mathcal{B}_f\big)$ are countably $\delta$-porous, quantitatively, with a computable porosity constant. When further metric and analytic assumptions are placed on $X$, $Y$, and $f$, our theorems generalize the well-known Bonk--Heinonen theorem and Sarvas' theorem to a large class of metric spaces. Moreover, our results are optimal in terms of the underlying geometric structures. As a direct application, we obtain the important V\"ais\"al\"a's inequality in greatest generality. Applying our main results to special cases, we solve an open problem of Heinonen--Rickman and an open question of Heinonen--Semmes.