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.