---
tag: EGAJPZ71VW
kind: concept
name: Quasiregular map
msc: [30C65]
sources:
  - url: https://en.wikipedia.org/wiki/Quasiregular_map
same_as:
  wikidata: Q7269522
  wikipedia: Quasiregular map
---

## Summary

A quasiregular map is a map between domains of $\R^n$ whose infinitesimal distortion is bounded.
It sends small balls to sets that are round up to a fixed factor.
Unlike a [[quasiconformal map|SB8KG46NHP]], it need not be injective, so it generalizes holomorphic functions rather than conformal maps.
In the plane, quasiregular maps are holomorphic functions composed with quasiconformal homeomorphisms.
In dimension $n \ge 3$ they give a rich substitute for holomorphic functions, since conformal maps there are only Möbius maps.

## Explanations

### L1 human

A holomorphic function stretches the plane near each point by the same amount in all directions.
A quasiregular map may stretch more in one direction than in another, but the ratio is bounded by a fixed constant at all points.
It may fold and cover some points several times, like $z \mapsto z^2$.
Results of Reshetnyak show that such maps, when not constant, still keep key features of holomorphic functions: they are open and their preimages of points are discrete.