A quasiregular map is a generalization of holomorphic functions to higher‑dimensional Euclidean spaces, extending the notion of quasiconformal mappings by allowing isolated branch points. Formally, let $n\ge 2$ and let $\Omega\subset \mathbb{R}^{n}$ be an open set. A continuous mapping $f:\Omega\rightarrow\mathbb{R}^{n}$ belongs to the Sobolev space $W_{\mathrm{loc}}^{1,n}(\Omega,\mathbb{R}^{n})$ and satisfies the distortion inequality
$$ |Df(x)|^{n}\le K, J_{f}(x) \qquad \text{for a.e. }x\in\Omega, $$
where $Df(x)$ is the differential (Jacobian matrix) of $f$ at $x$, $|Df(x)|$ denotes its operator norm, $J_{f}(x)=\det Df(x)$ is the Jacobian determinant, and $K\ge 1$ is a constant called the outer dilatation of $f$. The smallest such $K$ is denoted $K_{O}(f)$. Equivalently, one may require the inner dilatation $K_{I}(f)=\operatorname*{ess,sup}{x\in\Omega}\frac{|Df(x)|^{n}}{J{f}(x)}$ to be bounded, and the map is called $K$-quasiregular when both $K_{O}(f)$ and $K_{I}(f)$ are bounded by the same constant $K$.
Basic properties
- Continuity, openness, and discreteness – Every quasiregular map is continuous, open (images of open sets are open), and discrete (pre‑images of points consist of isolated points).
- Luzin N and N⁻¹ properties – Quasiregular maps send sets of measure zero to sets of measure zero (N‑property) and the pre‑image of a set of measure zero also has measure zero (N⁻¹‑property).
- Local Hölder continuity – A $K$-quasiregular map on $\Omega$ is locally Hölder continuous with exponent $\alpha = K^{1/(1-n)}$.
- Analytic representation – In dimension $n=2$, quasiregular maps coincide with the classical quasiconformal maps, and every planar quasiregular map can be written as the composition of a holomorphic function with a quasiconformal homeomorphism (Stoilow factorization).
Historical development
The concept was introduced by Yu. G. Reshetnyak in the mid‑1960s as part of his work on the theory of mappings with bounded distortion. Subsequent systematic study was carried out by J. Väisälä, O. Martio, S. Rickman, and others during the 1970s and 1980s, leading to a rich theory that parallels many classical results of complex analysis.
Key theorems
- Reshetnyak’s theorem – A quasiregular mapping is a branched covering; in particular, it is either a local homeomorphism or has isolated branch points of finite multiplicity.
- Rickman’s Picard theorem – A non‑constant $K$-quasiregular map $\mathbb{R}^{n}\to\mathbb{R}^{n}$ omits at most a finite set of points, with the cardinality bounded in terms of $n$ and $K$. This is a higher‑dimensional analogue of Picard’s theorem for entire holomorphic functions.
- Martio–Rickman–Väisälä theorem – The branch set of a quasiregular map in $\mathbb{R}^{n}$ has topological dimension at most $n-2$.
Typical examples
-
Power‑type maps: For $a>0$ and $x\in\mathbb{R}^{n}\setminus{0}$, the map
$$ f(x)=|x|^{a-1}x $$ is $K$-quasiregular with $K$ depending only on $a$ and $n$. When $a$ is an integer, this reduces to the classical polynomial mapping in complex analysis. -
Quasiconformal homeomorphisms: Any $K$-quasiconformal homeomorphism (i.e., a bijective quasiregular map) is quasiregular with the same dilatation constant.
-
Zorich map: The map $Z:\mathbb{R}^{n}\to\mathbb{R}^{n}$ defined by
$$ Z(x_{1},\dots ,x_{n}) = \bigl(e^{x_{1}}\cos x_{2},, e^{x_{1}}\sin x_{2},, x_{3},\dots ,x_{n}\bigr) $$ is a quasiregular covering of $\mathbb{R}^{n}\setminus{0}$.
Applications
- Geometric function theory – Quasiregular mappings provide a framework for extending classical complex‑analytic results (e.g., normal families, value distribution) to higher dimensions.
- Nonlinear potential theory – Their distortion control allows the development of capacity and modulus techniques analogous to those used for conformal maps.
- Geometric group theory – Quasiregular maps appear in the study of boundaries of hyperbolic groups, where they serve as models for quasi‑isometries of the hyperbolic space.
- Dynamics – Iteration of quasiregular self‑maps of $\mathbb{R}^{n}$ (or of the sphere $\mathbb{S}^{n}$) leads to a higher‑dimensional analogue of complex dynamics, including the definition of Julia and Fatou sets for quasiregular maps.
Related concepts
- Quasiconformal mapping – A quasiregular map that is additionally a homeomorphism.
- Finite distortion mapping – A broader class where the distortion inequality holds with a variable function $K(x)$ rather than a uniform constant.
- Branch set – The set of points where a quasiregular map fails to be locally injective; it has Hausdorff dimension at most $n-2$.
References (selected)
- Reshetnyak, Y. G. “Space Mappings with Bounded Distortion.” Siberian Mathematical Journal, vol. 6, 1965, pp. 1012–1020.
- Martio, O., Rickman, S., Väisälä, J. “Definitions for Quasiregular Mappings.” Ann. Acad. Sci. Fenn. Ser. A I Math., vol. 448, 1970, pp. 1–40.
- Rickman, S. “Quasiregular Mappings.” Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 26, Springer, 1993.
- Heinonen, J., Koskela, P. “Quasiconformal Maps in Metric Spaces.” Acta Math., vol. 181, 1998, pp. 1–61.
