arXiv2026
Given a continuous map $f:X\to Y$ from a continuum-connected space into a metric space, we introduce an associated canonical pullback metric $d_f$ on $X$ by combining the target diameter structure with the Mazurkiewicz construction as \[ d_f(x,x')=\inf\{\operatorname{diam}_Y f(E): E\subset X\text{ is a continuum containing }x,x'\}. \] Our main result shows that if $f:(X,ρ)\to Y$ is $L$-BDD, then $d_f$ is quantitatively equivalent to the Mazurkiewicz metric $ρ_M$, $(d_f)_M=d_f$, every continuum satisfies $\operatorname{diam}_{d_f}E=\operatorname{diam}_Y f(E)$ and $f:(X,d_f)\to Y$ is $1$-BDD and hence $1$-BLD. For bounded-turning spaces, this identifies a canonical bi-Lipschitz metric realization in the sense that every finite-to-one topological branched covering onto a locally length metric manifold admits such a compatible induced metric without any a priori BDD hypothesis. There are two main geometric applications of the induced pullback metric. Firstly, we prove an index-free quantitative porosity theorem for branch sets and their images at points of controlled linear or inverse linear dilatation, extending the well-known Bonk-Heinonen theorem and Sarvas theorem for branch set of Euclidean quasiregular mappings to a large class of metric spaces. Secondly, we prove that every finite-to-one BDD branched covering over a doubling bounded-turning target space admits a quantitative bi-Lipschitz embedding into the target times a finite-dimensional Euclidean space. As direct corollaries, we solve several well-known open problems asked by Heinonen, Rickman and Semmes.