An analogue of Green’s functions for quasiregular maps

The iteration theory of quadratic polynomials is a very well understood area of complex dynamics. Since the first fractal images of the Mandelbrot set appeared in the 1980s, and the work of Douady and Hubbard [6, 7] initiated a great deal of interest in this subject, there has been a consistent surge of interest. These fascinating images that arise from simply defined functions yields a surprisingly intricate theory that is still not completely settled. We refer to the texts of Beardon [1], Carleson and Gamelin [5], and Milnor [14] for introductions to complex dynamics.

The introduction of quasiconformal and quasiregular mappings into the theory of complex dynamics was another major reason for the impetus of the 1980s. Douady and Hubbard’s work on polynomial-like mappings [8] and Sullivan’s proof of the No Wandering Domains Theorem for rational maps [16] illustrated the utility of this approach. For more applications of quasiregular mappings in complex dynamics, we refer to the text of Branner and Fagella [4].

More recently, the study of the iteration theory of quasiregular mappings themselves has become an object of interest. This approach naturally extends complex dynamics into higher real dimensions. However, due to tools available only in two dimensions such as the Measurable Riemann Mapping Theorem and the Stoilow Decomposition Theorem, more can be said in the two dimensional case. For an introduction to quasiregular dynamics, we refer to Bergweiler’s survey [2] as a starting point.

The subject of our study is the class of quasiregular maps of the form

(1.1)

$$H_{K,\theta,c}(z)=\left[\left(\frac{K+1}{2}\right)z+e^{2i\theta}\left(\frac{K-1}{2}\right)\overline{z}\right]^{2}+c,$$

where $K>1$, $\theta\in(-\pi/2,\pi/2]$ and $c\in\mathbb{C}$. The dynamics of these mappings were first studied by the second named author and Goodman in [11], and subsequently by the second named author and Fryer in [9, 10]. We will review the important points from these papers in the preliminary section, but here let us point out that these are the simplest non-injective quasiregular mappings as they have constant complex dilatation

$$\mu_{H_{K,\theta,c}}\equiv e^{2i\theta}\left(\frac{K-1}{K+1}\right)\in\mathbb{D}.$$

It is evident that we can write

$$H_{K,\theta,c}=P_{c}\circ h_{K,\theta},$$

where $P_{c}(z)=z^{2}+c$ and $h_{K,\theta}$ is the affine map that stretches by a factor $K>1$ in the direction $e^{i\theta}$. This Stoilow decomposition of $H_{K,\theta,c}$ is also a characterization of degree two quasiregular maps in $\mathbb{C}$ with constant complex dilatation: every such map is conformally conjugate to one of the form (1.1). Observe that if $K=1$, then we return to the case of quadratic polynomials. One of the goals of this paper is to illustrate new phenomena that can occur in this setting, when compared to complex dynamics. Figure 1 gives an example where the function has a saddle fixed point, a situation that cannot occur for holomorphic functions.

We briefly review some of the well-known material from the dynamics of quadratic polynomials to set the stage for what is to come. We recall that every quadratic polynomial is linearly conjugate to one of the form $P_{c}(z)=z^{2}+c$.

Given a non-injective polynomial $P$, $\mathbb{C}$ can be decomposed into the escaping set

$$I(P)=\{z\in\mathbb{C}:P^{n}(z)\to\infty\}$$

and the bounded orbit set

$$BO(P)=\{z\in\mathbb{C}:\text{ there exists }M>0\text{ such that }|P^{n}(z)|\leq M\text{ for all }n\in\mathbb{N}\}.$$

The bounded orbit set is often called the filled Julia set in the literature, and denoted by $K(P)$, but we will reserve the use of $K$ for the maximal dilatation of quasiregular mappings.

The chaotic set is called the Julia set, and denoted by $J(P)$, whereas the stable set is called the Fatou set, and denoted by $F(P)$. Slightly more formally, the Fatou set is where the family of iterates locally forms a normal family, and the Julia set is the complement of the Fatou set. The importance of the escaping set is that the definition of the Julia set is a difficult one to check, but the fact that

$$J(P)=\partial I(P)$$

gives a highly useful way to visualize the Julia set. Critical fixed points of polynomials are called superattracting fixed points and are necessarily in the Fatou set. In particular, if we extend $P$ to the Riemann sphere $\mathbb{C}_{\infty}$, then we see that $\infty$ is a superattracting fixed point of $P$.

For quadratic polynomials, we have a dichotomy in the dynamical behavior. If $0\in BO(P)$, then both $BO(P)$ and $J(P)$ are connected. On the other hand, if $0\in I(P)$ then both $BO(P)$ and $J(P)$ are a Cantor set, that is, a totally disconnected, compact, perfect set. These distinctive behaviors give rise to the definition of the Mandelbrot set in parameter space:

$$\mathcal{M}=\{c\in\mathbb{C}:0\in BO(P_{c})\}.$$

Böttcher’s Theorem allows us to conformally conjugate near a superattracting fixed point of local index $d$ to the power map $z\mapsto z^{d}$. For quadratic polynomials, as $\infty$ is a superattracting fixed point, we may conformally conjugate to $z\mapsto z^{2}$ in a neighborhood of $\infty$. More precisely, there exists a conformal map $\varphi_{c}$ and a neighborhood $W_{c}$ of $\infty$ such that

(1.2)

$$\varphi_{c}(P_{c}(z))=\left[\varphi_{c}(z)\right]^{2}$$

for $z\in W_{c}$. It is desirable to extend (1.2) to the largest possible domain. The only obstruction to extending $W_{c}$ to larger domains is if the orbit of the critical point at $0$ escapes. Therefore, if $0\in\mathcal{M}$, then (1.2) holds on all of $I(P_{c})$, whereas if $0\notin\mathcal{M}$, then we cannot do so.

A resolution to this problem is to use harmonic Green’s functions. The real valued function

$$G_{c}(z)=\log|\varphi_{c}(z)|$$

satisfies the functional equation

(1.3)

$$G_{c}(P_{c}(z))=2G_{c}(z)$$

initially in $W_{c}$. This time the functional equation allows us to extend the domain of definition of $G_{c}$ to all of $I(P_{c})$, and it turns out that $G_{c}$ is precisely the Green’s function for the exterior domain of $BO(P_{c})$ with a pole at $\infty$. In particular, $G_{c}$ is harmonic and $G_{c}(z)\to 0$ as $\operatorname{dist}(z,J(P_{c}))\to 0$.

The equipotentials $G_{c}(z)=t$ give a particularly striking picture of the dynamics of $P_{c}$. If $c\in\mathcal{M}$, then these equipotentials give a foliation of $I(P_{c})$ through simple closed curves. On the other hand, if $c\notin\mathcal{M}$, then while the equipotentials for $t$ large enough are simple closed curves, there is a critical value given by $G_{c}(0)=t_{0}$ for which the level curve forms a figure-eight shape. For $0<t<t_{0}$, the level curves are then disconnected.

For mappings of the form (1.1), the plane can be again decomposed into the escaping set and the bounded orbit set.

In [9], an analogous result to Böttcher’s Theorem is established with gives the existence of a quasiconformal map $\varphi_{K,\theta,c}$ which conjugates $H_{K,\theta,c}$ to $H_{K,\theta,0}$ on a neighborhood $W_{K,\theta,c}$ of $\infty$ contained in the escaping set, that is,

$$\varphi_{K,\theta,c}(H_{K,\theta,c}(z))=H_{K,\theta,0}(\varphi_{K,\theta,c}(z))$$

for $z\in W_{K,\theta,c}$. The question then arises of whether this conjugation can be extended. Once again, the Mandelbrot set in parameter space plays an important role. For $K>1$ and $\theta\in(-\pi/2,\pi/2]$, we define the Mandelbrot set

$$\mathcal{M}_{K,\theta}=\{c\in\mathbb{C}:0\in BO(H_{K,\theta,c})\}.$$

We note that this time there is not an equivalent formulation in terms of the connectedness of the Julia set, as $J(H)$ and $\partial I(H)$ may differ for quasiregular maps, see for example [3, Example 7.3].

Returning to the Böttcher coordinate, it was shown in [9, Theorem 2.4] that if $c\in\mathcal{M}_{K,\theta}$, then $\varphi_{K,\theta,c}$ may be extended to a locally quasiconformal map on all of $I(H_{K,\theta,c})$, whereas if $c\notin\mathcal{M}_{K,\theta}$, then it cannot. Our first main result shows that the Green’s function idea also works in this setting.

We do not expect $G_{K,\theta,c}$ to be harmonic, although we leave the question of the regularity of this function to future work. However, $G_{K,\theta,c}$ still has equipotentials that are useful dynamically. For $t>0$, let us define

$$E(t)=E_{K,\theta,c}(t)=\{z\in\mathbb{C}:G_{K,\theta,c}(z)=t\},$$

and

$$U(t)=U_{K,\theta,c}(t)=\{z\in\mathbb{C}:G_{K,\theta,c}(z)>t\}.$$

Using these notions, we will give an alternative proof and mild refinement of [11, Theorem 5.3 and 5.4].

Recalling the dichotomy that $J(P_{c})$ is either connected or a Cantor set, we now turn to the question of what happens when $\partial I(H_{K,\theta,c})$ has uncountably many components. First, we give a condition that ensures that $\partial I(H_{K,\theta,c})$ is not a Cantor set. Recall that a periodic point of period $n$ for $f$ is a solution of $f^{n}(z)=z$. Note that a fixed point is considered a periodic point of all periods.

Finally, we aim to illustrate that the situation in Theorem 1.3 can indeed occur and exhibit some of the other features that can occur for the dynamics of these mappings that contrast with the dynamics of quadratic polynomials.

Figure 2 illustrates an example of the second case in Theorem 1.4, and Figure 1 illustrates an example of the third case.

The paper is organized as follows. In Section 2, we recall preliminary material on quasiregular mappings in the plane, and known material on the dynamics of the mappings $H_{K,\theta,c}$. In Section 3 we prove Theorem 1.1 by constructing our analogue of a Green’s function. In Section 4 we use equipotentials for our Green’s functions to establish Theorem 1.2. In Section 5 we prove Theorem 1.3. In Section 6, we give a classification of the type of fixed point of $H_{K,0,c}$ based on its location and finally in Section 7 we use the classification to help establish the examples in Theorem 1.4.

A quasiconformal mapping $f:\mathbb{C}\to\mathbb{C}$ is a homeomorphism such that $f$ is in the Sobolev space $W^{1}_{2,loc}(\mathbb{C})$ and there exists $k\in[0,1)$ such that the complex dilatation $\mu_{f}=f_{\overline{z}}/f_{z}$ satisfies

$$|\mu_{f}(z)|\leq k$$

almost everywhere in $\mathbb{C}$. The dilatation at $z\in\mathbb{C}$ is

$$K_{f}(z):=\frac{1+|\mu_{f}(z)|}{1-|\mu_{f}(z)|}.$$

A mapping is called $K$-quasiconformal if $K_{f}(z)\leq K$ almost everywhere. The smallest such constant is called the maximal dilatation and denoted by $K(f)$. If we drop the assumption on injectivity, then $f$ is called a quasiregular mapping. We refer to the books of Rickman [15] and Iwaniec and Martin [13] for much more on the development of the theory of quasiregular mappings.

In the plane, we have the following two crucial results. First, there is a surprising correspondence between quasiconformal mappings and measurable functions, see for example [13, p.8].

Moreover, every quasiregular mapping has an important decomposition, see for example [13, p.254].

In fact, in the plane, some sources use this decomposition as the definition of a quasiregular mapping. However, this approach does not generalize to higher dimensions, so even though this paper is two dimensional, we will keep Stoilow Decomposition as a theorem.

Here we review the properties of the mappings we will focus on in this paper. Let $K>1$ and $\theta\in(-\pi/2,\pi/2]$. Then we define $h_{K,\theta}$ to be the stretch by factor $K$ in the direction of $e^{i\theta}$. Evidently, if $\theta=0$, then

$$h_{K,0}(x+iy)=Kx+iy.$$

Interpreting $h_{K,\theta}$ in terms of $h_{K,0}$, if $\rho_{\theta}$ denotes the rotation counter-clockwise by angle $\theta$, then we see that

$$h_{K,\theta}=\rho_{\theta}\circ h_{K,0}\circ\rho_{-\theta}.$$

From this we can obtain the following explicit formulas for $h_{K,\theta}$:

	$\displaystyle h_{K,\theta}(z)$	$\displaystyle=\left(\frac{K+1}{2}\right)z+e^{2i\theta}\left(\frac{K-1}{2}\right)\overline{z}$
		$\displaystyle=\left[x(K\cos^{2}(\theta)+\sin^{2}(\theta))+y(K-1)\sin(\theta)\cos(\theta)\right]$
		$\displaystyle\hskip 36.135pt+i\left[x(K-1)\cos(\theta)\sin(\theta)+y(K\sin^{2}(\theta)+\cos^{2}(\theta))\right].$

It is clear that

$$\mu_{h_{K,\theta}}\equiv e^{2i\theta}\left(\frac{K-1}{K+1}\right),$$

and it follows from the Measurable Riemann Mapping Theorem and [11, Proposition 3.1] that any quasiconformal mapping of $\mathbb{C}$ with constant complex dilatation given as above arises by conjugating $h_{K,\theta}$ by a complex linear map.

If we view quasiconformal mappings with constant complex dilatation as the simplest in their class, then we can view quadratic polynomials as the simplest non-injective holomorphic functions. In light of the Stoilow Decomposition Theorem, the simplest quasiregular mappings arise as compositions of quadratic polynomials and mappings of the form $h_{K,\theta}$. As defined in the introduction, for $K>1$, $\theta\in(-\pi/2,\pi/2]$ and $c\in\mathbb{C}$, we have $H_{K,\theta,c}=P_{c}\circ h_{K,\theta}$. For later use, we note that if $r>0$, then

(2.1)

$$h_{K,\theta}(rz)=rh_{K,\theta}(z)\text{ and }H_{K,\theta,0}(rz)=r^{2}H_{K,\theta,0}(z).$$

It is worth remarking on the domains of $K$ and $\theta$. Evidently a stretch in the direction $e^{i\theta}$ is the same as a stretch in the direction $e^{i(\theta+\pi)}$, and so we only need consider $\theta\in(-\pi/2,\pi/2]$. A stretch by factor $K>1$ in the direction $e^{i\theta}$ is conjugate to a contraction by factor $1/K$ in the direction $e^{i(\theta+\pi/2)}$. We could thus restrict the domain of $\theta$ further and allow any positive value for $K$, but we will usually use the convention that $K>1$ instead. There may be occasion to conveniently allow $K>0$, but we will alert the reader when this is the case. We recall from [11, Proposition 3.1] that every degree two quasiregular mapping of the plane with constant complex dilatation is linearly conjugate to $H_{K,\theta,c}$ for some choice of parameters.

As $H_{K,\theta,c}$ is quasiregular, we may consider the behavior of its iterates. In the plane, every uniformly quasiregular mapping, that is, one for which there is a uniform bound on the maximal dilatation of the iterates, is known to be a quasiconformal conjugate of a holomorphic map. This means that the dynamics of such mappings yields essentially nothing new when compared to the features of complex dynamics. Importantly, from the point of view of independent interest, the mappings $H_{K,\theta,c}$ are not uniformly quasiregular [10, Theorem 1.12].

The escaping set for $H_{K,\theta,c}$ is a non-empty, open neighborhood of $\infty$ by [11, Theorem 4.3]. It follows that the bounded orbit set is the complement of the escaping set. All of $I(H_{K,\theta,c}),BO(H_{K,\theta,c})$ and $\partial I(H_{K,\theta,c})$ are completely invariant under $H_{K,\theta,c}$.

When $c=0$, we can guarantee a neighborhood of $0$ is in $BO(H_{K,\theta,0})$.

We also have control of the growth of $H_{K,\theta,c}$ near infinity.

As this lemma also illustrates, the escaping set of $H_{K,\theta,c}$ is non-empty, and we can therefore ask for a conjugation to a simpler mapping in a neighborhood of infinity, analogous to Böttcher’s Theorem. The following result yields this.

Finally, we will need the following version of the Riemann-Hurwitz formula. Due to the Stoilow Decomposition Theorem, there are no technical issues when applying the Riemann-Hurwitz formula with quasiregular mappings instead of holomorphic functions, but we refer to [11, Corollary 5.2] for more details.