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In the mathematical field of [[mathematical analysis|analysis]], '''quasiregular maps''' are a class of continuous maps between Euclidean spaces '''R'''''n'' of the same dimension or, more generally, between [[Riemannian manifold]]s of the same dimension, which share some of the
basic properties with [[holomorphic function]]s of one complex variable.

==Motivation==

The theory of holomorphic (=[[analytic function|analytic]])
functions of one complex variable is
one of the most beautiful and most useful parts of the whole mathematics.

One drawback of his theory is that it deals only with maps between
two-dimensional spaces ([[Riemann surfaces]]). The theory of functions
of several complex variables has a different character, mainly because
analytic functions of several variabes are not [[conformal map|conformal]].
Conformal maps can be defined between Euclidean spaces of arbitrary dimension,
but when the dimension is greater than 2, this class of maps is very small:
it consists of [[Möbius transformations]] only.
This is a theorem of [[Joseph Liouville]]; relaxing the smoothness
assumptions does not help, as proved by [[Yurii Reshetnyak]].<ref name="resh">
{{cite book|author=Yu. G. Reshetnyak|
title=Stability theorems in geometry and analysis|
publisher=[[Kluwer Academic Publishers|Kluwer]]|year=1994}}</ref>

This suggests the search of a generalization of the property of conformality which
would give a rich and interesting class of maps in higher dimension.

==Definition==

A [[differentiable map]] ''f'' of a region ''D'' in '''R'''''n''
to '''R'''''n'' is called ''K''-quasiregular if the following inequality holds at all points in ''D'':

:<math> \| Df(x)\|^n\leq K|J_f(x)| \,</math>.

Here ''K'' ≥ 1 is a constant, ''J''''f'' is the [[Jacobian determinant]], ''Df'' is the derivative, that is the linear map defined by the [[Jacobian matrix and determinant|Jacobi matrix]], and ||·|| is the usual (Euclidean) [[matrix norm|norm]] of the matrix.

The development of the theory of such maps showed that it is unreasonable to restrict oneself to differentiable maps in the classical sense, and that the "correct" class of maps consists of continuous maps in the [[Sobolev space]] ''W''{{su|p=1,''n''|b=loc}} whose partial derivatives
in the sense of [[distribution (mathematics)|distributions]] have locally summable ''n''-th power, and such that the above inequality is satisfied [[almost everywhere]]. This is a formal definition
of a ''K''-quasiregular map. A map is called '''quasiregular''' if it is ''K''-quasiregular with some ''K''. Constant maps are excluded from the class of quasiregular maps.

==Properties==

The fundamental theorem about quasiregular maps was proved by
Reshetnyak:<ref name="resh2" />

:''Quasiregular maps are open and discrete''.

This means that the images of [[open set]]s are open and that preimages of
points consist of isolated points. In dimension 2, these two properties give a
topological characterization of the class of non-constant analytic functions:
every continuous open and discrete map of a plane domain to the plane can be pre-composed
with a [[homeomorphism]], so that the result is an analytic function.
This is a theorem of [[Simion Stoilov]].

Reshetnyak's teorem implies that all pure topological results about
analytic functions (such that the Maximum Modulus Principle, Rouché's theorem
etc.) extend to quasiregular maps.

Injective quasiregular maps are called [[quasiconformal]].
A simple example of non-injective quasiregular map is given in
cylindrical coordinated in 3-space by the formula

:<math> (r,\theta,z)\mapsto (r,2\theta,z). </math>

This map is 2-quasiregular. It is smooth everywhere except the ''z''-axis.
A remarkable fact is that all smooth quasiregular maps are local
homeomorphisms. Even more remarkable is that every quasiregular local homeomorphism
'''R'''''n'' → '''R'''''n'', where ''n'' ≥ 3,
is a homeomorphism (this is a [[Zorich's theorem|theorem of Vladimir Zorich]]<ref name="resh2">
{{cite book|author=Yu. G. Reshetnyak|
title=Space mappings with bounded distortion|
publisher=[[American Mathematical Society]]|
year=1989}}</ref>).

This explains why in the definition of quasiregular maps it is not reasonable
to restrict oneself to smooth maps: all smooth quasiregular maps of '''R'''''n'' to
itself are quasiconformal.

==Rickman's theorem==

Many theorems about geometric properties of holomorphic functions of one complex variable have been extended to quasiregular maps. These extensions are usually highly non-trivial.

Perhaps the most famous result of this sort is the extension of [[Picard's theorem]] which is due to Seppo Rickman:<ref>{{cite book|author=S. Rickman|title=Quasiregular mappings|publisher=[[Springer Verlag]]|year=1993}}</ref>

:''A K-quasiregular map'' '''R'''''n'' → '''R'''''n'' ''can omit at most a finite set''.

When ''n'' = 2, this omitted set can contain at most two points (this is a simple
extension of Picard's theorem). But when ''n'' > 2, the omitted set can contain
more than two points, and its cardinality can be estimated from above in terms of ''n'' and ''K''.

==Connection with potential theory==

If ''f'' is an analytic function, then log ''|f|'' is [[subharmonic function|subharmonic]],
and [[harmonic function|harmonic]] away from the zeros of ''f''. The corresponding fact
for quasiregular maps is that log ''|f|'' satisfies a certain non-linear
[[partial differential equation]] of [[elliptic operator|elliptic type]].
This discovery of Reshetnyak stimulated the development of
'''non-linear potential theory''', which treats this kind of equations
as the usual [[potential theory]] treats harmonic and subharmonic functions.

==References==

<references/>

==External links==
* Yurii Reshetnyak on the Russian Wikipedia [[:ru:Решетняк]]

* Vladimir Zorich on the Russian Wikipedia [[:ru:Зорич, Владимир Антонович]]

[[Category:Mathematical analysis]]

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