Polynomial-Type Quasiregular Maps

• Polynomial-type quasiregular maps are finite-degree self-maps that extend to the sphere Sⁿ, exhibiting polynomial-like behavior at infinity.
• They are characterized by an interplay between algebraic degree and geometric distortion, with conditions like deg f > K_I(f) guiding their dynamics.
• These maps serve as a higher-dimensional analogue to complex polynomials, influencing studies of escaping sets, parameter spaces, and commutation properties.

Polynomial-type quasiregular maps are quasiregular self-maps $f:\mathbb{R}^n\to\mathbb{R}^n$ characterized by the condition

$\lim_{x\to\infty}|f(x)|=\infty.$

Equivalently, they are the finite-degree members of the class of entire quasiregular maps. In higher-dimensional dynamics they play the role of polynomials in one complex variable: they extend to quasiregular self-maps of the one-point compactification $S^n$, admit a topological degree $\deg f$, and their dynamics at infinity are organized by the interaction between algebraic degree and geometric distortion, especially the thresholds $\deg f>K_I(f)$ and, in commutation problems, $\deg f>K(f)$ (Bergweiler et al., 2012, Fletcher et al., 2014, Tsantaris, 2019).

1. Definitions and structural position

A continuous map $f\in W^{1,n}_{\mathrm{loc}}(\Omega,\mathbb{R}^n)$ is quasiregular if there exists $K_O\ge 1$ such that

$|Df(x)|^n \le K_O\,J_f(x)\qquad\text{a.e.},$

or equivalently $K_I\ge 1$ such that

$\lim_{x\to\infty}|f(x)|=\infty.$0

where $\lim_{x\to\infty}|f(x)|=\infty.$1 is the Jacobian determinant and $\lim_{x\to\infty}|f(x)|=\infty.$2 is the minimal stretch. The maximal dilatation is

$\lim_{x\to\infty}|f(x)|=\infty.$3

A nonconstant quasiregular map is open and discrete (Bergweiler et al., 2012, Tsantaris, 2019).

For entire quasiregular maps $\lim_{x\to\infty}|f(x)|=\infty.$4, the dichotomy between polynomial type and transcendental type is defined by behavior at infinity. The map is of polynomial type when $\lim_{x\to\infty}|f(x)|=\infty.$5, and of transcendental type when this limit does not exist. In the polynomial-type case the degree is finite, and one may equivalently write

$\lim_{x\to\infty}|f(x)|=\infty.$6

This class is structurally distinguished because polynomial-type maps extend to quasiregular self-maps of the one-point compactification $\lim_{x\to\infty}|f(x)|=\infty.$7, so they fall under the previously developed Fatou–Julia theory for quasiregular self-maps of the sphere (Bergweiler et al., 2012, Fletcher et al., 2014).

That compactifiability is one of the main reasons polynomial-type dynamics is closer to rational or polynomial dynamics than the transcendental theory. The transcendental case requires new machinery such as capacity-based Julia sets, escaping sets, and the pits-effect dichotomy, whereas the polynomial-type case is already anchored to the sphere model and to finite-degree dynamics (Bergweiler et al., 2012).

2. Infinity as a strongly superattracting point

A central device in the theory is inversion

$\lim_{x\to\infty}|f(x)|=\infty.$8

which conjugates infinity to the origin. If $\lim_{x\to\infty}|f(x)|=\infty.$9 is of polynomial type and

$S^n$0

then $S^n$1 is quasiregular near $S^n$2, and

$S^n$3

If

$S^n$4

then $S^n$5 is a strongly superattracting fixed point of $S^n$6 in the sense that $S^n$7 (Fletcher et al., 2014).

The local engine behind this theory is a shell-distortion estimate. For

$S^n$8

there exist $S^n$9 and $\deg f$0 such that for $\deg f$1 and $\deg f$2,

$\deg f$3

where

$\deg f$4

After inversion, this becomes a quantitative control on the growth of polynomial-type maps at infinity (Fletcher et al., 2014).

Several global consequences follow when $\deg f$5. If $\deg f$6, then for every $\deg f$7,

$\deg f$8

For the escaping set

$\deg f$9

and the fast escaping set

$\deg f>K_I(f)$0

one has

$\deg f>K_I(f)$1

Moreover, escaping points have comparable rates: for $\deg f>K_I(f)$2, there exists $\deg f>K_I(f)$3 such that for large $\deg f>K_I(f)$4,

$\deg f>K_I(f)$5

and on any compact $\deg f>K_I(f)$6, there exists $\deg f>K_I(f)$7 such that for large $\deg f>K_I(f)$8,

$\deg f>K_I(f)$9

Thus, under $\deg f>K(f)$0, infinity behaves as the global analogue of a strongly superattracting fixed point (Fletcher et al., 2014).

This degree condition also interfaces with the sphere-based Julia theory. Earlier work summarized in the transcendental iteration paper states that if $\deg f>K(f)$1 is of polynomial type and

$\deg f>K(f)$2

then $\deg f>K(f)$3 and many classical properties hold; under this hypothesis the capacity-based Julia definition agrees with the classical one for uniformly quasiregular maps, and in particular for polynomials (Bergweiler et al., 2012).

3. Degree, dilatation, and commutation

Polynomial-type quasiregular maps enter a rigidity theory for commuting maps. If $\deg f>K(f)$4 are permutable,

$\deg f>K(f)$5

and $\deg f>K(f)$6 is of polynomial type with

$\deg f>K(f)$7

then $\deg f>K(f)$8 must also be of polynomial type. Equivalently, a polynomial-type quasiregular map can commute with a transcendental-type quasiregular map only if

$\deg f>K(f)$9

This is presented as a higher-dimensional analogue of a theorem of Baker and Iyer from holomorphic dynamics (Tsantaris, 2019).

The proof is a growth comparison argument. For polynomial-type $f\in W^{1,n}_{\mathrm{loc}}(\Omega,\mathbb{R}^n)$0, Rickman-theoretic estimates give, for large $f\in W^{1,n}_{\mathrm{loc}}(\Omega,\mathbb{R}^n)$1,

$f\in W^{1,n}_{\mathrm{loc}}(\Omega,\mathbb{R}^n)$2

with

$f\in W^{1,n}_{\mathrm{loc}}(\Omega,\mathbb{R}^n)$3

If $f\in W^{1,n}_{\mathrm{loc}}(\Omega,\mathbb{R}^n)$4, then $f\in W^{1,n}_{\mathrm{loc}}(\Omega,\mathbb{R}^n)$5. Under the commutation relation one has an upper bound

$f\in W^{1,n}_{\mathrm{loc}}(\Omega,\mathbb{R}^n)$6

while a Pólya-type lemma yields

$f\in W^{1,n}_{\mathrm{loc}}(\Omega,\mathbb{R}^n)$7

and hence

$f\in W^{1,n}_{\mathrm{loc}}(\Omega,\mathbb{R}^n)$8

for large $f\in W^{1,n}_{\mathrm{loc}}(\Omega,\mathbb{R}^n)$9. Comparing these forces a bounded-growth-ratio property that contradicts the transcendental-growth lemma used in the paper. Therefore a transcendental-type map cannot commute with such a $K_O\ge 1$0 (Tsantaris, 2019).

The condition $K_O\ge 1$1 is described there as natural in quasiregular dynamics, playing the role that $K_O\ge 1$2 plays in holomorphic dynamics. In this sense, polynomial-type maps whose degree exceeds their dilatation are too expansive, in the quasiregular sense relevant to the theory, to commute with transcendental-type maps (Tsantaris, 2019).

4. Canonical planar families and parameter spaces

In the plane, a foundational model class is obtained by composing a polynomial with an affine stretch. The basic stretch is

$K_O\ge 1$3

whose complex dilatation is constant: $K_O\ge 1$4 Every composition of a quadratic polynomial and such an affine stretch is linearly conjugate to

$K_O\ge 1$5

and the paper introduces the classes

$K_O\ge 1$6

with $K_O\ge 1$7 (Fletcher et al., 2010).

These are polynomial-type quasiregular maps in $K_O\ge 1$8. For $K_O\ge 1$9, with $|Df(x)|^n \le K_O\,J_f(x)\qquad\text{a.e.},$0 a polynomial of degree $|Df(x)|^n \le K_O\,J_f(x)\qquad\text{a.e.},$1 and $|Df(x)|^n \le K_O\,J_f(x)\qquad\text{a.e.},$2 $|Df(x)|^n \le K_O\,J_f(x)\qquad\text{a.e.},$3-bi-Lipschitz, the escaping set

$|Df(x)|^n \le K_O\,J_f(x)\qquad\text{a.e.},$4

is non-empty and open, and $|Df(x)|^n \le K_O\,J_f(x)\qquad\text{a.e.},$5 is perfect. In the $|Df(x)|^n \le K_O\,J_f(x)\qquad\text{a.e.},$6 family, the branch set is

$|Df(x)|^n \le K_O\,J_f(x)\qquad\text{a.e.},$7

and the bounded-orbit set

$|Df(x)|^n \le K_O\,J_f(x)\qquad\text{a.e.},$8

plays the role of the filled Julia set. The connectedness criterion is sharp: $|Df(x)|^n \le K_O\,J_f(x)\qquad\text{a.e.},$9 If $K_I\ge 1$0 contains the branch set, then $K_I\ge 1$1 is infinitely connected. This is a marked contrast with classical quadratic polynomials, where escape of the critical orbit produces a totally disconnected filled Julia set (Fletcher et al., 2010).

For fixed stretch parameter, the associated Mandelbrot-type set is

$K_I\ge 1$2

It satisfies

$K_I\ge 1$3

is compact, and obeys

$K_I\ge 1$4

For $K_I\ge 1$5,

$K_I\ge 1$6

This reproduces the critical-orbit philosophy of the classical Mandelbrot set, but with the branch point $K_I\ge 1$7 replacing the critical point of a quadratic polynomial (Fletcher et al., 2010).

A Böttcher-type linearization holds near infinity for the same quadratic-stretch family. If $K_I\ge 1$8 is affine and $K_I\ge 1$9, then there exists a neighborhood $\lim_{x\to\infty}|f(x)|=\infty.$00 of infinity and a quasiconformal map $\lim_{x\to\infty}|f(x)|=\infty.$01 such that

$\lim_{x\to\infty}|f(x)|=\infty.$02

Equivalently,

$\lim_{x\to\infty}|f(x)|=\infty.$03

The coordinate $\lim_{x\to\infty}|f(x)|=\infty.$04 is asymptotically conformal at infinity: $\lim_{x\to\infty}|f(x)|=\infty.$05 The same work proves that $\lim_{x\to\infty}|f(x)|=\infty.$06 is not uniformly quasiregular, so these maps remain genuinely quasiregular rather than collapsing to the uniformly quasiregular/rational regime (Fletcher et al., 2012).

5. Linearization, Green-type functions, and intrinsic geometry

For degree-two planar quasiregular maps with constant complex dilatation, a particularly explicit normal form is

$\lim_{x\to\infty}|f(x)|=\infty.$07

The paper on Green-function analogues states that every degree-two planar quasiregular map with constant complex dilatation is linearly conjugate to one of these $\lim_{x\to\infty}|f(x)|=\infty.$08. The associated escaping and bounded-orbit sets are

$\lim_{x\to\infty}|f(x)|=\infty.$09

and

$\lim_{x\to\infty}|f(x)|=\infty.$10

with $\lim_{x\to\infty}|f(x)|=\infty.$11, both completely invariant (Broderius et al., 2024).

For the unperturbed map $\lim_{x\to\infty}|f(x)|=\infty.$12, the scaling relation

$\lim_{x\to\infty}|f(x)|=\infty.$13

implies that each ray $\lim_{x\to\infty}|f(x)|=\infty.$14 meets $\lim_{x\to\infty}|f(x)|=\infty.$15 in exactly one point $\lim_{x\to\infty}|f(x)|=\infty.$16. This defines a radial coordinate

$\lim_{x\to\infty}|f(x)|=\infty.$17

satisfying

$\lim_{x\to\infty}|f(x)|=\infty.$18

Using the quasiconformal Böttcher-type coordinate $\lim_{x\to\infty}|f(x)|=\infty.$19 near infinity for general $\lim_{x\to\infty}|f(x)|=\infty.$20,

$\lim_{x\to\infty}|f(x)|=\infty.$21

one obtains

$\lim_{x\to\infty}|f(x)|=\infty.$22

The analogue Green’s function is then

$\lim_{x\to\infty}|f(x)|=\infty.$23

and extends to all of $\lim_{x\to\infty}|f(x)|=\infty.$24 as a nonnegative continuous function that is identically zero on $\lim_{x\to\infty}|f(x)|=\infty.$25, positive on $\lim_{x\to\infty}|f(x)|=\infty.$26, and satisfies

$\lim_{x\to\infty}|f(x)|=\infty.$27

everywhere (Broderius et al., 2024).

This function reproduces the dynamical role of the polynomial Green’s function without its harmonic regularity. Equipotentials

$\lim_{x\to\infty}|f(x)|=\infty.$28

satisfy

$\lim_{x\to\infty}|f(x)|=\infty.$29

where $\lim_{x\to\infty}|f(x)|=\infty.$30. For sufficiently large $\lim_{x\to\infty}|f(x)|=\infty.$31, $\lim_{x\to\infty}|f(x)|=\infty.$32 is a simple closed curve; in general, $\lim_{x\to\infty}|f(x)|=\infty.$33 is a finite union of closed curves. The topology of $\lim_{x\to\infty}|f(x)|=\infty.$34 depends sharply on the parameter. If $\lim_{x\to\infty}|f(x)|=\infty.$35, then $\lim_{x\to\infty}|f(x)|=\infty.$36 is connected. If $\lim_{x\to\infty}|f(x)|=\infty.$37, then $\lim_{x\to\infty}|f(x)|=\infty.$38 has uncountably many components, and for $\lim_{x\to\infty}|f(x)|=\infty.$39, the level set $\lim_{x\to\infty}|f(x)|=\infty.$40 has $\lim_{x\to\infty}|f(x)|=\infty.$41 components where

$\lim_{x\to\infty}|f(x)|=\infty.$42

The same family exhibits several phenomena absent from quadratic polynomial dynamics: up to four fixed points, attracting fixed points outside the Mandelbrot-type set, and saddle fixed points with curved local stable manifolds (Broderius et al., 2024).

A related geometric development introduces the intrinsic metric

$\lim_{x\to\infty}|f(x)|=\infty.$43

on $\lim_{x\to\infty}|f(x)|=\infty.$44, where $\lim_{x\to\infty}|f(x)|=\infty.$45 is the hyperbolic metric. For a non-constant $\lim_{x\to\infty}|f(x)|=\infty.$46-quasiregular self-map $\lim_{x\to\infty}|f(x)|=\infty.$47,

$\lim_{x\to\infty}|f(x)|=\infty.$48

That paper explicitly describes the bound $\lim_{x\to\infty}|f(x)|=\infty.$49 as the key polynomial-type feature of the distortion theory (Fujimura et al., 2020).

6. Flexibility of construction and higher-dimensional examples

Despite the rigidity created by degree and dilatation, polynomial-type quasiregular maps are also remarkably flexible. A striking example is the realization theorem for maximum modulus sets. For

$\lim_{x\to\infty}|f(x)|=\infty.$50

the paper on maximum modulus sets proves that if $\lim_{x\to\infty}|f(x)|=\infty.$51 and $\lim_{x\to\infty}|f(x)|=\infty.$52 is closed and meets every sphere centered at the origin, then for each $\lim_{x\to\infty}|f(x)|=\infty.$53 there exists a quasiregular map $\lim_{x\to\infty}|f(x)|=\infty.$54 of polynomial type and degree $\lim_{x\to\infty}|f(x)|=\infty.$55 such that

$\lim_{x\to\infty}|f(x)|=\infty.$56

The construction uses a Zorich map $\lim_{x\to\infty}|f(x)|=\infty.$57, quasiregular power mappings $\lim_{x\to\infty}|f(x)|=\infty.$58 satisfying

$\lim_{x\to\infty}|f(x)|=\infty.$59

with

$\lim_{x\to\infty}|f(x)|=\infty.$60

and a quasiconformal deformation $\lim_{x\to\infty}|f(x)|=\infty.$61 that is the identity on $\lim_{x\to\infty}|f(x)|=\infty.$62 and strictly decreases radius off $\lim_{x\to\infty}|f(x)|=\infty.$63. The final map is

$\lim_{x\to\infty}|f(x)|=\infty.$64

This gives an exact realization theorem unavailable in the classical entire-function setting (Fletcher et al., 2020).

A second family of constructions shows that polynomial-type maps in $\lim_{x\to\infty}|f(x)|=\infty.$65 can support highly nonclassical global dynamics. The interpolation theorem of the round-ring paper constructs, for each $\lim_{x\to\infty}|f(x)|=\infty.$66, a quasiregular map

$\lim_{x\to\infty}|f(x)|=\infty.$67

such that

$\lim_{x\to\infty}|f(x)|=\infty.$68

with dilatation independent of $\lim_{x\to\infty}|f(x)|=\infty.$69. By assembling such interpolations with degrees $\lim_{x\to\infty}|f(x)|=\infty.$70, the authors build a polynomial-type quasiregular map $\lim_{x\to\infty}|f(x)|=\infty.$71 whose quasi-Fatou set contains wandering components, whose quasi-Fatou components are bounded and hollow, and whose Julia set has components that are genuine round spheres. In particular, there exist sequences of spheres $\lim_{x\to\infty}|f(x)|=\infty.$72 and hollow components $\lim_{x\to\infty}|f(x)|=\infty.$73 such that

$\lim_{x\to\infty}|f(x)|=\infty.$74

and $\lim_{x\to\infty}|f(x)|=\infty.$75 has $\lim_{x\to\infty}|f(x)|=\infty.$76 and $\lim_{x\to\infty}|f(x)|=\infty.$77 as boundary components. The same construction can be tuned so that the resulting polynomial-type map grows as quickly, or as slowly, as desired along subsequences (Burkart et al., 2024).

Taken together, these results show that polynomial-type quasiregular maps are simultaneously constrained and expansive as a research domain. They are constrained by finite degree, compactifiability to $\lim_{x\to\infty}|f(x)|=\infty.$78, and degree–dilatation thresholds that govern Julia sets, escape rates, and commutation. They are expansive in the sense that they admit quasiconformal Böttcher coordinates, Green-type dynamical potentials, highly nonclassical parameter spaces in planar model families, arbitrary prescribed maximum modulus sets, and higher-dimensional examples with bounded hollow quasi-Fatou components and spherical Julia components (Fletcher et al., 2014, Tsantaris, 2019, Fletcher et al., 2010, Fletcher et al., 2012, Broderius et al., 2024, Fletcher et al., 2020, Burkart et al., 2024).