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. Let f : X → Y be a continuous, discrete and open mapping. Let B_f be the branch set of f , i.e. the set consisting of points in X at which f fails to be a local homeomorphism.
In this paper, addressing Heinonen’s ICM 02 talk, we study the geometry of the branch set B_f of a quasiregular mapping between metric n-manifolds. In particular, we show that B_f ∩ {x ∈ X : H_f (x) < ∞} is countably porous, as is its image f(B_f) ∩ {x ∈ X :H_f(x) < ∞} . As a corollary, B_f ∩ {x ∈ X : H_f (x) < ∞} and its image are null sets with respect to any locally doubling measures on X and Y , respectively. Moreover, if
either H_f (x) ≤ H or H_f *(x) ≤ H for all x ∈ X, then both B_f and f(B_f) are countably δ-porous, quantitatively, with a computable porosity constant.
Both our methods and results are new even in the Euclidean space . Our methods are greatly inspired by the recent developments from quantitative topology and allow us to formulate our results even without the cohomology assumptions on the underlying
metric spaces; our results give a general decomposition of the branch set of a discrete open mapping according to the control on the linear dilatation.
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 Väisälä’s inequality in greatest generality.
Applying our main results to special cases, we solve a recent conjecture of Fassler et al., an open problem of Heinonen–Rickman, and an open question of Heinonen–Semmes.