The Canonical Induced Metric for Topological Branched Coverings and Geometric Applications

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 $$d_f(x,x^{\mathrm{\prime }})=\inf \{{\mathrm{diam}}_{Y}f(E):E\subset X\text{ is a continuum containing }x,x^{\mathrm{\prime }}\}\mathrm{.}$$ Our main result shows that if $f:(X,\rho )\to Y$ is $L$-BDD, then $d_f$ is quantitatively equivalent to the Mazurkiewicz metric ${\rho }_M$, $(d_f{)}_M=d_f$, every continuum satisfies ${\mathrm{diam}}_{d_f}E={\mathrm{diam}}_{Y}f(E)$ 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.