--- title: Polynomial-Type Quasiregular Maps url: https://www.emergentmind.com/topics/polynomial-type-quasiregular-maps type: topic --- # Polynomial-Type Quasiregular Maps 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, \(\deg f>K(f)\) [1210.3972, 1404.2778, 1912.04152]. ## 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 \[ J_f(x)\le K_I\,l(Df(x))^n\qquad\text{a.e.}, \] where \(J_f\) is the Jacobian determinant and \(l(Df(x))\) is the minimal stretch. The maximal dilatation is \[ K(f)=\max\{K_O(f),K_I(f)\}. \] A nonconstant quasiregular map is open and discrete [1210.3972, 1912.04152]. For entire quasiregular maps \(f:\mathbb{R}^n\to\mathbb{R}^n\), the dichotomy between polynomial type and transcendental type is defined by behavior at infinity. The map is of polynomial type when \(\lim_{x\to\infty}|f(x)|=\infty\), and of transcendental type when this limit does not exist. In the polynomial-type case the degree is finite, and one may equivalently write \[ \deg f=\sup_{y\in\mathbb{R}^n}\#f^{-1}(y)<\infty. \] This class is structurally distinguished because polynomial-type maps extend to quasiregular self-maps of the one-point compactification \(S^n\), so they fall under the previously developed Fatou–Julia theory for quasiregular self-maps of the sphere [1210.3972, 1404.2778]. 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 [1210.3972]. ## 2. Infinity as a strongly superattracting point A central device in the theory is inversion \[ g(x)=\frac{x}{|x|^2},\qquad g=g^{-1}, \] which conjugates infinity to the origin. If \(f\) is of polynomial type and \[ \tilde f=g\circ f\circ g, \] then \(\tilde f\) is quasiregular near \(0\), and \[ i(0,\tilde f)=i(\infty,f)=\deg f. \] If \[ \deg f>K_I(f), \] then \(0\) is a strongly superattracting fixed point of \(\tilde f\) in the sense that \(i(0,\tilde f)>K_I(\tilde f)\) [1404.2778]. The local engine behind this theory is a shell-distortion estimate. For \[ l(x,r)=\inf_{|y-x|=r}|f(y)-f(x)|,\qquad L(x,r)=\sup_{|y-x|=r}|f(y)-f(x)|, \] there exist \(C>1\) and \(r_0>0\) such that for \(0K_I(f)\). If \(M(r,f)=\sup_{|x|=r}|f(x)|\), then for every \(S>1\), \[ \limsup_{r\to\infty}\frac{M(Sr,f)}{M(r,f)}<\infty. \] For the escaping set \[ I(f)=\{x\in\mathbb{R}^n:f^k(x)\to\infty\}, \] and the fast escaping set \[ A(f)=\{x\in\mathbb{R}^n:\exists L\in\mathbb{N},\ |f^{k+L}(x)|>M_k(R,f)\ \forall k\}, \] one has \[ I(f)=A(f). \] Moreover, escaping points have comparable rates: for \(x,y\in I(f)\), there exists \(N\) such that for large \(k\), \[ |f^k(x)|<|f^{k+N}(y)|, \] and on any compact \(F\subset I(f)\), there exists \(a>1\) such that for large \(k\), \[ \frac1a\le \frac{\log|f^k(x)|}{\log|f^k(y)|}\le a \qquad (x,y\in F). \] Thus, under \(\deg f>K_I(f)\), infinity behaves as the global analogue of a strongly superattracting fixed point [1404.2778]. This degree condition also interfaces with the sphere-based Julia theory. Earlier work summarized in the transcendental iteration paper states that if \(f\) is of polynomial type and \[ \deg(f)>K_I(f), \] then \(J(f)\neq\emptyset\) 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 [1210.3972]. ## 3. Degree, dilatation, and commutation Polynomial-type quasiregular maps enter a rigidity theory for commuting maps. If \(f,g:\mathbb{R}^d\to\mathbb{R}^d\) are permutable, \[ f\circ g=g\circ f, \] and \(g\) is of polynomial type with \[ \deg g>K(g), \] then \(f\) must also be of polynomial type. Equivalently, a polynomial-type quasiregular map can commute with a transcendental-type quasiregular map only if \[ \deg g\le K(g). \] This is presented as a higher-dimensional analogue of a theorem of Baker and Iyer from holomorphic dynamics [1912.04152]. The proof is a growth comparison argument. For polynomial-type \(g\), Rickman-theoretic estimates give, for large \(|x|\), \[ A|x|^{n_1}\le |g(x)|\le B|x|^{n_2}, \] with \[ n_1=\left(\frac{\deg g}{K_I(g)}\right)^{1/(d-1)},\qquad n_2=\left(\deg g\,K_O(g)\right)^{1/(d-1)}. \] If \(\deg g>K(g)\), then \(n_1>1\). Under the commutation relation one has an upper bound \[ M(r,g\circ f)\le B\,M(r,f)^{n_2}, \] while a Pólya-type lemma yields \[ M(r,f\circ g)=M(r,g\circ f)\ge M\!\left(c\,M(r/2,g),f\right), \] and hence \[ M(r,g\circ f)\ge M\!\left(cA\left(\frac r2\right)^{n_1},f\right) \] for large \(r\). 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 \(g\) [1912.04152]. The condition \(\deg g>K(g)\) is described there as natural in quasiregular dynamics, playing the role that \(\deg g\ge 2\) 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 [1912.04152]. ## 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 \[ h_{K,\theta}(z)=\left(\frac{K+1}{2}\right)z+e^{2i\theta}\left(\frac{K-1}{2}\right)\overline z, \] whose complex dilatation is constant: \[ \mu_{h_{K,\theta}}=e^{2i\theta}\frac{K-1}{K+1}. \] Every composition of a quadratic polynomial and such an affine stretch is linearly conjugate to \[ f_{K,\theta,C}(z)=h_{K,\theta}(z)^2+C, \] and the paper introduces the classes \[ QA=\{f=g\circ h,\ g(z)=z^2+c,\ h=h_{K,\theta}\}, \qquad PA=\{f=g\circ h,\ g \text{ polynomial of degree }\ge 2,\ h=h_{K,\theta}\}, \] with \(QA\subset PA\) [1006.0622]. These are polynomial-type quasiregular maps in \(\mathbb{R}^2\). For \(f=g\circ h\), with \(g\) a polynomial of degree \(d\ge 2\) and \(h\) \(L\)-bi-Lipschitz, the escaping set \[ I(f)=\{z\in\mathbb{C}:f^n(z)\to\infty\} \] is non-empty and open, and \(\partial I(f)\) is perfect. In the \(QA\) family, the branch set is \[ B(f)=\{0\}, \] and the bounded-orbit set \[ N(f)=\{z\in\mathbb{C}:\{f^n(z)\}\text{ is bounded}\} \] plays the role of the filled Julia set. The connectedness criterion is sharp: \[ N(f)\text{ is connected if and only if }I(f)\cap B(f)=\emptyset. \] If \(I(f)\) contains the branch set, then \(N(f)\) 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 [1006.0622]. For fixed stretch parameter, the associated Mandelbrot-type set is \[ \mathcal{M}_{K,\theta}=\{c\in\mathbb{C}:0\notin I(f_c)\}, \qquad f_c(z)=h_{K,\theta}(z)^2+c. \] It satisfies \[ \mathcal{M}_{K,\theta}=\{c\in\mathbb{C}:\partial I(f_c)\text{ is connected}\}, \] is compact, and obeys \[ \mathcal{M}_{K,\theta}\subset \{c\in\mathbb{C}:|c|\le 2L_1^{-2}\}, \qquad L_1=\min\{K,1\}. \] For \(\theta=0\), \[ \mathcal{M}_{K,0}\cap\mathbb{R}= \left[-\frac{2}{K^2},\,\frac{1}{4K^2}\right]. \] This reproduces the critical-orbit philosophy of the classical Mandelbrot set, but with the branch point \(0\) replacing the critical point of a quadratic polynomial [1006.0622]. A Böttcher-type linearization holds near infinity for the same quadratic-stretch family. If \(h\) is affine and \(f(z)=h(z)^2+c\), then there exists a neighborhood \(U\) of infinity and a quasiconformal map \(\psi\) such that \[ h(\psi(z))^2=\psi(f(z)) \qquad (z\in U). \] Equivalently, \[ \psi\circ f=H\circ\psi,\qquad H(z)=h(z)^2. \] The coordinate \(\psi\) is asymptotically conformal at infinity: \[ |\mu_\psi(z)|\to 0\qquad\text{as }|z|\to\infty. \] The same work proves that \(h(z)^2+c\) is not uniformly quasiregular, so these maps remain genuinely quasiregular rather than collapsing to the uniformly quasiregular/rational regime [1205.1978]. ## 5. Linearization, Green-type functions, and intrinsic geometry For degree-two planar quasiregular maps with constant complex dilatation, a particularly explicit normal form is \[ H_{K,\theta,c}(z)=\left[\left(\frac{K+1}{2}\right)z +e^{2i\theta}\left(\frac{K-1}{2}\right)\overline z\right]^2+c =\bigl(h_{K,\theta}(z)\bigr)^2+c. \] 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 \(H_{K,\theta,c}\). The associated escaping and bounded-orbit sets are \[ I(H_{K,\theta,c})=\{z:H_{K,\theta,c}^n(z)\to\infty\}, \] and \[ BO(H_{K,\theta,c})=\{z:\exists M>0\text{ such that }|H_{K,\theta,c}^n(z)|\le M\ \forall n\}, \] with \(BO(H_{K,\theta,c})=\mathbb{C}\setminus I(H_{K,\theta,c})\), both completely invariant [2408.11256]. For the unperturbed map \(H_{K,\theta,0}\), the scaling relation \[ H_{K,\theta,0}(rz)=r^2H_{K,\theta,0}(z) \] implies that each ray \(R_\phi\) meets \(\partial I(H_{K,\theta,0})\) in exactly one point \(b_{K,\theta}(\phi)\). This defines a radial coordinate \[ \tau_{K,\theta,0}(z)=\frac{z}{b_{K,\theta}(\arg z)}, \] satisfying \[ \tau_{K,\theta,0}(H_{K,\theta,0}(z)) =\bigl[\tau_{K,\theta,0}(z)\bigr]^2. \] Using the quasiconformal Böttcher-type coordinate \(\varphi_{K,\theta,c}\) near infinity for general \(c\), \[ \varphi_{K,\theta,c}\circ H_{K,\theta,c} =H_{K,\theta,0}\circ \varphi_{K,\theta,c}, \] one obtains \[ \tau_{K,\theta,c}(H_{K,\theta,c}(z)) =[\tau_{K,\theta,c}(z)]^2. \] The analogue Green’s function is then \[ G_{K,\theta,c}(z)=\log\tau_{K,\theta,c}(z), \] and extends to all of \(\mathbb{C}\) as a nonnegative continuous function that is identically zero on \(BO(H_{K,\theta,c})\), positive on \(I(H_{K,\theta,c})\), and satisfies \[ G_{K,\theta,c}(H_{K,\theta,c}(z))=2G_{K,\theta,c}(z) \] everywhere [2408.11256]. This function reproduces the dynamical role of the polynomial Green’s function without its harmonic regularity. Equipotentials \[ E_{K,\theta,c}(t)=\{z:G_{K,\theta,c}(z)=t\} \] satisfy \[ E(2^nt)=H_{K,\theta,c}^n(E(t)), \qquad E(t)=\partial U(t), \] where \(U(t)=\{z:G_{K,\theta,c}(z)>t\}\). For sufficiently large \(t\), \(E(t)\) is a simple closed curve; in general, \(E(t)\) is a finite union of closed curves. The topology of \(\partial I(H_{K,\theta,c})\) depends sharply on the parameter. If \(c\in\mathcal{M}_{K,\theta}\), then \(\partial I(H_{K,\theta,c})\) is connected. If \(c\notin\mathcal{M}_{K,\theta}\), then \(\partial I(H_{K,\theta,c})\) has uncountably many components, and for \(t