2.1. Conformal metrics

Suppose X is a domain in Sⁿ. A conformal metric on X is a continuous form $\tau(x) |{\rm d}x|$ for which τ is strictly positive. A conformal metric induces a distance function given by

\begin{equation*} d(u,v) = \inf \int_{\gamma} \tau (x) |{\rm d}x|,\end{equation*}

where the infimum is over all locally rectifiable paths in X joining u and v. We will often write τ_X and d_X for the metric and distance function on X, as long as the context is clear.

We remark that not every distance function arises in this way.

An important point to consider is whether the metrics are complete, that is, whether Cauchy sequences in the corresponding distance function converge. For example, the hyperbolic metric is complete on $\mathbb{B}^n$, whereas the Euclidean metric is not. However, every proper subdomain of Sⁿ carries a complete conformal metric, namely the quasihyperbolic metric. It is sometimes convenient to assume that metrics are complete and then appeal to the following lemma.

The proof of this lemma is the same as that of [Reference Beardon and Minda1, Lemma 2.1]. We recall that d ₁ and d ₂ are bi-Lipschitz equivalent distance functions on X if there exists $L\geq 1$ such that

\begin{equation*} \frac{d_2(x,y)}{L} \leq d_1 (x,y) \leq Ld_2(x,y),\end{equation*}

for all $x,y\in X$. The smallest such L for which the above inequalities hold is called the isometric distortion.

2.2. ω-Continuity

The notion of ω-continuity is a quantitative way of describing continuity and is closely related to uniform continuity. Throughout, X and Y are subdomains of Sⁿ with distance functions d_X and d_Y, respectively, arising from conformal metrics.

We recall that $C(X,Y)$ is the space of continuous maps $f:X\to Y$, where continuity is with respect to d_X and d_Y. The space $C(X,Y)$ carries a distance function defined as follows (c.f. [Reference Beardon and Minda1, §3]): let $(K_k)_{k=1}^{\infty}$ be a compact exhaustion of X and for $f,g \in C(X,Y)$, set

\begin{equation*} d_k(f,g) = \sup \{d_Y( f(x), g(x) ):x\in K_k \}, \end{equation*}

and

\begin{equation*} d_{X,Y}(f,g) = \sum_{k=1}^{\infty} \frac{1}{2^k} \left ( \frac{d_k(f,g) }{1+d_k(f,g)} \right ).\end{equation*}

Then, $d_{X,Y}(f_m,f) \to 0$ if and only if $f_m\to f$ uniformly on compact subsets of X. The topology arising from this distance function is called the topology of uniform convergence on compact sets, or the topology of local uniform convergence, and does not depend on the particular choice of compact exhaustion. Consequently, when we say $f_m\to f$ in $C(X,Y)$, this means local uniform convergence relative to d_X and d_Y or, equivalently, convergence relative to the distance $d_{X,Y}$ on $C(X,Y)$. Note also that the topology induced on $C(X,Y)$ is not changed when the distance functions d_X and d_Y are replaced with topologically equivalent distances.

We will often want to talk about a family $\mathcal{F}$ of functions in $C(X,Y)$ being relatively compact. This means that the closure of $\mathcal{F}$ in $C(X,Y)$ is compact. Since sequential compactness coincides with compactness in metric spaces, we are free to use either notion when discussing compact sets in $C(X,Y)$. In particular, $\mathcal{F}$ is relatively compact if and only if any sequence of functions $f_m \in \mathcal{F}$ is guaranteed to have a subsequence which converges uniformly on compact subsets to an element of $C(X,Y)$.

We now recall the notion of ω-continuity.

Definition 2.3. Let $\omega:[0,\infty) \to [0,\infty)$ be a continuous increasing function with $w(0) = 0$. We call a map $f:X\to Y$ ω-continuous if f has modulus of continuity ω, that is, if

\begin{equation*} d_Y (f(x),f(y)) \leq \omega ( d_X(x,y)),\end{equation*}

for all $x,y\in X$.

We will show that uniform ω-continuity on compact subsets and local uniform ω-continuity are the same notions with respect to conformal metrics.

Proof. Each point in X has a compact neighbourhood, so uniformly ω-continuous on compact subsets implies locally uniformly ω-continuous.

For the converse, suppose that $\mathcal{F}$ is locally uniformly ω-continuous but not uniformly ω-continuous on compact sets. Then, there exist a compact set $E\subset X$, points $x_m, y_m \in E$ and $f_m\in\mathcal{F}$ such that, for all $m\in\mathbb{N}$,

(2.2)\begin{equation} d_Y (f_m(x_m),f_m(y_m)) \gt m \omega (d_X(x_m,y_m) ). \end{equation}

We may assume that the sequences $(x_m)$ and $(y_m)$ converge to limits x and y in E, respectively. We must have that $x\ne y$; otherwise, we obtain a contradiction from the local uniform ω-continuity of $\mathcal{F}$ at x. Using local uniform ω-continuity at x and y, it now follows from (2.2) that

(2.3)\begin{equation} d_Y (f_m(x),f_m(y)) \to \infty, \quad \mbox{as } m\to\infty. \end{equation}

Let $\gamma: [0,1] \to X$ be a path from x to y. For each point $u\in\gamma([0,1]),$ there exist $0 \lt r_u \lt (1/2)$ and $L_u \gt 0$ such that for all $p,q\in B_X(u,r_u)$ and all $f\in\mathcal{F}$,

\begin{equation*} d_Y (f(p),f(q)) \leq L_u \omega (d_X(p,q) ) \le L_u\omega(1). \end{equation*}

Since $\gamma([0,1])$ is compact, we can cover it with finitely many such balls, say $B_X(u_i,r_{u_i})$ for $i=1,\ldots, J$. It follows that, for all $f\in\mathcal{F}$,

\begin{equation*} d_Y(f(x), f(y)) \le \sum_{i=1}^J L_{u_i}\omega(1), \end{equation*}

in contradiction to (2.3).

2.3. Relative compactness and the Arzela–Ascoli Theorem

We recall the definition of equicontinuity.

The Arzela–Ascoli Theorem in this setting takes the form [Reference Munkres17, Theorem 47.1]:

If X and Y are subdomains of Sⁿ with conformal metrics, then they satisfy the hypotheses of the theorem. We can use the Arzela–Ascoli Theorem to formulate a characterization of relative compactness in the setting of ω-continuity, c.f. [Reference Beardon and Minda1, Theorem 7.1], where the completeness of d_Y shows that only one orbit needs to be checked for relative compactness.

We remark that the completeness of d_Y is necessary in Theorem 2.9, as the next example shows.

Example 2.10. Let $X=Y = (-1,1)^2$ equipped with the Euclidean distance. For $m\geq 2$ and $x\in (-1,1)$, consider the piecewise linear maps:

\begin{equation*} p_m(x) = \left \{\begin{array}{cc} x & x\in (-1,0], \\ \frac{2(m-1)x}{m} & x\in (0,1/2), \\ \frac{2x+m-2}{m} & x\in [1/2 ,1), \end{array} \right .\end{equation*}

and define $f_m(x,y) = (p_m(x) ,y )$ for $(x,y) \in X$. The family $\{f_m : m\geq 2 \} \subset C(X,Y)$ is uniformly Lipschitz, but the orbit of $(0,0)$ is relatively compact in Y and the orbit of $(1/2,0)$ is not.

2.4. The Escher condition

The idea is that near $\partial Y$, disks of radius R in d_Y are contained in small disks in d_Z. It is worth bearing in mind the example of the hyperbolic metric in $\mathbb{B}^n$: near $\partial \mathbb{B}^n$, hyperbolic disks of radius 1 have small Euclidean diameter.

Theorem 2.12. Suppose that τ_Y is a complete metric on Y and $Y\subset Z \subset S^n$ with metric τ_Z. Then τ_Y satisfies an Escher condition relative to τ_Z if and only if

\begin{equation*} \lim_{x\to \zeta} \frac{\tau_Z(x)}{\tau_Y(x)} = 0,\end{equation*}

for all $\zeta \in \partial Y$.

The proof is identical to [Reference Beardon and Minda1, Theorem 10.1] and is omitted. The classical definition of a normal family of holomorphic functions in $\mathbb{C}$ requires the limit of a convergent subsequence to either be holomorphic or for the subsequence to diverge to infinity uniformly on compact subsets. The latter situation should be viewed in the context of the Riemann sphere and the subsequence converging uniformly on compact subsets to the point at infinity. In light of this, we recall the definition of a normal family from [Reference Beardon and Minda1, § 9].

In the discussion preceding this definition, X and Y are $\mathbb{C}$ with the Euclidean metric and Z is the Riemann sphere with the spherical metric. We remark that the specific choice of conformal metrics on $X,Y$ and Z is irrelevant in Definition 2.13, since any choices of conformal metrics generate the same topology on $C(X,Y)$ and $C(X,Z)$. Moreover, in the case that $\overline{Y} \subset Z$, the choice of Z does not matter in this definition. In this paper, we only consider the case when $\overline{Y} \subset Z$, often because we choose Z to be Sⁿ.

We next show how, for locally uniformly ω-continuous families, the Escher condition allows us to use relative compactness and normality as synonyms, c.f. [Reference Beardon and Minda1, Theorem 11.1].

Proof. The normality of $\mathcal{F}$ relative to Z implies that $\mathcal{F}$ is relatively compact in $C(X,Z)$.

For the converse, suppose that $\mathcal{F}$ is relatively compact in $C(X,Z)$. Let $f_m \in \mathcal{F}$ be a sequence and, passing to a subsequence and relabelling if necessary, we can assume that $f_m \to f$ locally uniformly in $C(X,Z)$. We need to show that either f is in $C(X,Y)$ or f is constant. Let us assume that f is not in $C(X,Y)$, in which case there exists $x_0 \in X$ with $f(x_0) = \lim_{m\to \infty} f_m(x_0) \in \partial Y$. Take $x\in X$ and ϵ > 0. Since $\{x_0,x\}$ is compact and $\mathcal{F}$ is locally uniformly ω-continuous, Proposition 2.6 implies that

(2.4)\begin{equation} d_Y(f_m(x),f_m(x_0)) \leq L \omega (d_X(x,x_0)), \end{equation}

for all $m\in \mathbb{N}$. Set $C = L\omega(d_X(x,x_0))$. The Escher condition means that we can find a compact set $E\subset Y$ such that if $v\in Y\setminus E$ and $d_Y(u,v) \lt C$, then

(2.5)\begin{equation} d_Z(u,v) \lt \frac{\epsilon}{C}d_Y(u,v). \end{equation}

Since E is a compact subset of Y and $f_m(x_0)\to f(x_0)\in\partial Y$, there is $M\in \mathbb{N}$ such that if $m\geq M$, then $f_m(x_0)\in Y\setminus E$ and $d_Z(f_m(x_0) , f(x_0)) \lt \epsilon$. Hence, by (2.4) and (2.5), if $m\geq M$, then

\begin{equation*} d_Z(f_m(x) , f(x_0)) \leq d_Z(f_m(x) , f_m(x_0)) + d_Z(f_m(x_0),f(x_0)) \lt 2\epsilon.\end{equation*}

It follows that $f(x) = f(x_0)$, and hence f is a constant map into $\partial Y$.

Note that, unlike [Reference Beardon and Minda1, Theorem 11.1], the above result does not assume that the metrics on the codomains Y and Z are complete. Theorem 2.14 immediately has the following corollary.

3.1. Preliminaries

A quasiregular mapping in a domain $U\subset \mathbb{R}^n$ for $n\geq 2$ is a continuous mapping in the Sobolev space $W^1_{n,\textrm{loc}}(U)$ where there is a uniform bound on the distortion, that is, there exists $K\geq 1$ such that

\begin{equation*}|f^{\prime}(x)|^n \leq KJ_f(x),\end{equation*}

almost everywhere in U, where $|f^{\prime}(x)|$ is the operator norm of $f^{\prime}(x)$ and $J_f(x)$ is the Jacobian determinant of f at x. The minimal K for which this inequality holds is called the outer dilatation and is denoted by $K_O(f)$. As a consequence of this, there is also $K^{\prime} \geq 1$ such that

\begin{equation*}J_f(x) \leq K^{\prime} \inf_{|h|=1}|f^{\prime}(x)h|^n\end{equation*}

holds almost everywhere in U. The minimal K ^ʹ for which this inequality holds is called the inner dilatation and denoted by $K_I(f)$. We then have that $K(f)= \max \{K_O(f), K_I(f) \}$ is the maximal dilatation of f. A K-quasiregular mapping is a quasiregular mapping for which $K(f) \leq K$. An injective quasiregular mapping is called quasiconformal. It turns out that non-constant quasiregular mappings are discrete and open.

To deal either with quasiregular mappings that are defined at infinity or with quasiregular maps that have a discrete set of poles, we may pre- or post-compose by a Möbius map A (which could be chosen to be a spherical isometry) that moves the point at infinity to the origin and then locally apply the above condition. Such mappings are sometimes called quasimeromorphic, but we will keep the nomenclature quasiregular and bear in mind that the domain and range can include the point at infinity.

The main property that we are interested in here is the fact that quasiregular mappings are locally Hölder continuous, as the following result illustrates.

Theorem 3.1. ([Reference Rickman23], Theorem III.1.11)

Let $n\geq 2$ and $K\geq 1$. Let $U \subset \mathbb{R}^n$ be a bounded domain and $f:U \to \mathbb{R}^n$ be a bounded K-quasiregular mapping. Suppose that $E\subset U$ is compact, let $\delta = d(E, \partial U)$, where d denotes Euclidean distance and let r denote the diameter of f(U). Then

\begin{equation*} |f(x) - f(y)| \leq C |x-y|^{\alpha},\end{equation*}

for all $x\in E$ and $y \in U$, and where $\alpha = K^{1/(1-n)}$, $C = \lambda(n)\delta ^{-\alpha}r$ with $\lambda(n)$ a constant that depends only on n.

Henceforth, we will denote by α the constant $K^{1/(1-n)}$. Below, we will prove some results on the order of growth of various classes of mappings. We recall the required notions here. Suppose f: $U \to S^n$ is a non-constant K-quasiregular mapping. The local index $i(x,f)$ is the infimum of $\sup_y \operatorname{card}f^{-1}(y)\cap V$, where V runs over all neighbourhoods of x. For $y\in S^n$ and a Borel set E with $\overline{E}$ compact in U, define

\begin{equation*} n(E,y) = \sum_{x\in f^{-1}(y) \cap E} i(x,f),\end{equation*}

that is, $n(E,y)$ counts the number of points in $f^{-1}(y)\cap E$ with multiplicity. For r > 0, $n(r,y)$ is defined to be $n(\overline{B(0,r)},y)$.

We define $A_{f,\sigma}(r)$ to be the average of $n(r,y)$ over Sⁿ with respect to spherical measure, that is,

\begin{equation*} A_{f,\sigma}(r) = \frac{1}{\omega_n} \int_{\mathbb{R}^n} \frac{n(r,y)}{(1+|y|^2)^n } \:\,{\rm d}m(y) = \frac{1}{\omega_n} \int_{\overline{B(0,r)}} \frac{J_f(x)}{(1 + |f(x)|^2)^n} \,{\rm d}m(x),\end{equation*}

where ${\rm d}m$ denotes Lebesgue measure and ω_n the surface area of Sⁿ. If the context is clear, we abbreviate $A_{f,\sigma}(r)$ to A(r). We note that our formulas differ from those in Rickman’s monograph [Reference Rickman23] by a factor of 2ⁿ as we are identifying $\mathbb{R}^n \cup \{\infty \}$ with the unit sphere Sⁿ in $\mathbb{R}^{n+1}$, whereas Rickman uses a sphere of radius $1/2$. We also define

(3.1)\begin{equation} A(x_0,r) = \frac{1}{\omega_n} \int_{\overline{B(x_0,r)}} \frac{J_f(x)}{(1 + |f(x)|^2)^n} \: {\rm d}m(x), \end{equation}

which has the interpretation of the normalized spherical volume of $f(\overline{B(x_0,r)})$ with multiplicity taken into account.

If Y is an $(n-1)$-dimensional sphere in $\overline{\mathbb{R}^n}$, then $\nu(E,Y)$ denotes the average of $n(E,y)$ over Y with respect to the $(n-1)$-dimensional spherical measure. In the particular case where $E = \overline{B(0,r)}$, we write $\nu(r,s)$ for $\nu ( \overline{B(0,r)}, \partial B(0,s))$.

For quasiregular mappings on $\mathbb{R}^n$, the order µ_f and lower order λ_f of f are defined by

\begin{equation*} \mu_f = \limsup_{r\to \infty} \frac{\log A(r) }{\log r}, \qquad \lambda_f = \liminf_{r\to \infty} \frac{\log A(r)}{\log r}.\end{equation*}

If f has no poles, then the order and lower order can be expressed in terms of the maximum modulus function

\begin{equation*} M(r,f) = \sup_{|x| = r} |f(x)| \end{equation*}

via

\begin{equation*} \mu_f = \limsup_{r\to \infty} (n-1) \frac{\log\, \log M(r,f)}{\log r}, \qquad \lambda_f = \liminf_{r\to \infty} (n-1) \frac{\log\, \log M(r,f)}{\log r}.\end{equation*}

For quasiregular mappings in the unit ball $\mathbb{B}^n$, analogous definitions for order and lower order hold, with $\log r$ replaced by $\log({1}/({1-r}))$ and limits as $r\to 1^-$.

3.2. Families of K-quasiregular mappings

We now make precise the notion of a normal family of quasiregular mappings. First we need the following definition.

The reason for this definition is that if $X,Y\subset \mathbb{R}^n$ are domains and $f_m\in \mathcal{Q}_K(X,Y)$ is a sequence of K-quasiregular mappings that converges locally uniformly to a map $f\in C(X,Y)$, then f itself must be in $\mathcal{Q}_K(X,Y)$ by [Reference Rickman23, Theorem VI.8.6]. Extending to include the point at infinity in domain or range can be handled via Möbius maps. In the absence of such a uniform bound on the maximal dilatation of the mappings in the family, there is no longer a guarantee that any limit function is quasiregular. Consider, for example, the planar family $\{f_K(x+iy) = Kx+iy : K \gt 0 \}$.

Note that (iii) is the definition stated by Miniowitz in [Reference Miniowitz16]. Recall that statements (i) and (ii) are independent of the choice of conformal metrics on $X, Y$ and Sⁿ and that (ii) is equivalent to (iii); see the remarks preceding Definition 2.3. To see that (i) and (ii) are equivalent, we need to show that for $\mathcal{F}\subset \mathcal{Q}_K(X,Y)$, any map in the closure of $\mathcal{F}$ in $C(X,S^n)$ is either in the closure of $\mathcal{F}$ in $C(X,Y)$ or is a constant map into $\partial Y$. This can be obtained from the following quasiregular version of Hurwitz’s Theorem, which implies that a map in the closure of $\mathcal{Q}_K(X,Y)$ in $C(X,S^n)$ must be constant if its image contains any value $a\notin Y$.

There are various results in the literature describing when a family of quasiregular mappings is normal. The following version of Montel’s Theorem for quasiregular mappings has found many applications.

We remark that the statement of [Reference Miniowitz16, Theorem 5] is stronger, in that the omitted points are allowed to vary but must stay at least a fixed spherical distance apart. We also recall Miniowitz’s version of Zalcman’s Lemma, see [Reference Miniowitz16, Lemma 1].

Lemma 3.6. A family $\mathcal{F}$ of K-quasiregular mappings from $\mathbb{B}^n$ to Sⁿ is not normal at x ₀ if and only if there exist sequences $x_m\to x_0$, $f_m \in \mathcal{F}$ and positive reals $\rho_m \to 0^+$ such that

\begin{equation*} f_m(x_m + \rho_mw) \to g(w),\end{equation*}

locally uniformly on compact subsets of $\mathbb{R}^n$, where g is a non-constant quasiregular mapping $g:\mathbb{R}^n \to S^n$.

Here, a family $\mathcal{F}$ is said to be normal at x₀ if there exists a neighbourhood U of x ₀ such that the family of restrictions $\mathcal{F} |_U$ is normal. We remark that Miniowitz’s statement does not include the requirement that $x_m \to x_0$, but this can easily be achieved, see for example, [Reference Iwaniec and Martin10, 19.7.3].

The main result of this section is the following characterization of relative compactness for families of K-quasiregular mappings based on ω-continuity.

Proof. The sufficiency follows from Theorem 2.9. For the other direction, by Theorem 2.9, it suffices to prove that if $\mathcal{F}$ is relatively compact in $C(X,Y)$, then $\mathcal{F}$ is locally uniformly ω-continuous, with $\omega(t) = t^{\alpha}$. Our main tool is Theorem 3.1, but we need mappings on bounded Euclidean domains to be able to apply it.

To that end, let $x_0 \in X$, and choose s > 0 such that the closed ball $B=\overline{B_X(x_0, s)}$ is compact. Then, since B is compact and $\mathcal{F}$ is relatively compact in the topology of local uniform convergence on $C(X,Y)$, it is not hard to see that any sequence in $\bigcup_{f\in \mathcal{F}} f(B)$ has a subsequence that converges in Y. Hence, $\bigcup_{f\in \mathcal{F}} f(B)$ is relatively compact in Y. We conclude that d_Y and the spherical distance σ are bi-Lipschitz equivalent on $\bigcup_{f\in\mathcal{F}} f(B)$ with, say, isometric distortion L.

Now for $0 \lt r_0 \lt s/2$, set $E = \overline{B_X(x_0,r_0)}$ and $U = B_X(x_0,2r_0)$. Since $\mathcal{F}$ is relatively compact in $C(X,Y)$, the Arzela–Ascoli Theorem implies that $\mathcal{F}$ is equicontinuous on $\overline{U}$. Taking $\epsilon =\pi/(2L)$, we can choose r ₀ sufficiently small that

\begin{equation*} \operatorname{diam} _Y f(\overline{U}) \leq \epsilon,\end{equation*}

for all $f\in \mathcal{F}$. Hence,

\begin{equation*} \operatorname{diam}_{\sigma} f(\overline{U}) \leq L\epsilon = \pi/2,\end{equation*}

for all $f\in \mathcal{F}$. We also choose r ₀ small enough that $\operatorname{diam}_{\sigma} \overline{U} \leq \pi/2$.

Note that d_X and σ are bi-Lipschitz equivalent on $\overline{U}$ with, say, isometric distortion L ₁. Let A ₁: $S^n \to S^n$ be a spherical isometry which maps x ₀ to 0, and let $A_f:S^n\to S^n$ be a spherical isometry which maps $f(x_0)$ to 0. Then $A_1(\overline{U})$ and $A_f(f(\overline{U}))$ are contained in the closed unit ball $\overline{\mathbb{B}^n}$ for all $f\in \mathcal{F}$. Recalling (2.1), for $u,v\in\overline{\mathbb{B}^n}$, the Euclidean and spherical distances are related by $|u-v|\le \sigma(u,v)\le 2|u-v|$. For $f\in\mathcal{F}$, we then obtain, via Theorem 3.1 applied to $A_f\circ f \circ A_1^{-1}$, the domain $A_1(U)$ and compact set $A_1(E)$, the following chain of inequalities for $x,y\in E$:

\begin{align*} d_Y( f(x),f(y)) &\leq L\sigma (f(x) , f(y) ) \\ &= L \sigma ( A_f(f(x)) , A_f(f(y)) ) \\ &\leq 2L| A_f(f(x)) - A_f(f(y)) | \\ &\leq 2LC|A_1(x)-A_1(y)|^{\alpha}\\ &\leq 2LC \sigma (A_1(x) , A_1(y))^{\alpha}\\ &= 2LC \sigma (x,y)^{\alpha}\\ &\leq 2LCL_1^{\alpha} d_X(x,y)^{\alpha}. \end{align*}

By Theorem 3.1, C here depends only on $n,K$, the Euclidean distance $d(A_1(E) $, $\partial A_1(U) )$ and the Euclidean diameter of $A_f(f(U))$. Since $A_f(f(U))$ is contained in the closed unit ball, this diameter is bounded above by 2. The rest of the terms here are independent of the choice of f, and hence C can be chosen not to depend on f. Since x ₀ was arbitrary, we conclude that $\mathcal{F}$ is locally uniformly w-continuous on X.

The final statement of the theorem follows since if Y is Sⁿ, then (ii) always holds, and, in this case, normality and relative compactness coincide.

This characterization of normality of families of K-quasiregular mappings using the spherical distance σ in the domain and range allows us to recover the following result of Miniowitz.

Theorem 3.8. (Theorem 1, [Reference Miniowitz16])

Let $\mathcal{F}\subset \mathcal{Q}_K(X,S^n)$ be a family of K-quasiregular mappings on a domain $X\subset S^n$ for $n\geq 2$ equipped with the spherical distance σ. Then, $\mathcal{F}$ is a normal family if and only if for each compact subset $E\subset X$ there exists L > 0 such that

\begin{equation*} \sigma (f(x),f(y)) \leq L \omega ( \sigma (x,y) ),\end{equation*}

for all $x,y \in E$ and $f\in \mathcal{F}$, where $\omega(t) = t^{\alpha}$ and $\alpha = K^{1/(1-n)}$.

In the case where the domain X is the unit ball with the hyperbolic metric, we have the following result, which is presumably known to the experts.

Theorem 3.9. Let $\mathcal{F}\subset \mathcal{Q}_K(\mathbb{B}^n,S^n)$ be a family of K-quasiregular mappings defined on $\mathbb{B}^n$ for $n\geq 2$. Then $\mathcal{F}$ is a normal family if and only if for each compact subset $E\subset \mathbb{B}^n$ there exists L > 0 such that

\begin{equation*} \sigma (f(x),f(y)) \leq L \omega ( \rho( x,y) ),\end{equation*}

for all $x,y\in E$ and $f\in \mathcal{F}$, where ρ denotes the hyperbolic distance on $\mathbb{B}^n$ and $\omega(t) = t^{\alpha}$ with $\alpha = K^{1/(1-n)}$.

In the case where the range is $\mathbb{R}^n$ with the Euclidean metric, by the discussions at the start of this section, a family $\mathcal{F} \subset \mathcal{Q}_K(X,\mathbb{R}^n)$ is normal if and only if every sequence in $\mathcal{F}$ has a subsequence which either converges to an element of $\mathcal{Q}_K(X,\mathbb{R}^n)$ or diverges to infinity. In the normal family literature, a family is called finitely normal if the second case is ruled out, that is, if $\mathcal{F}$ is relatively compact in $C(X,\mathbb{R}^n)$. Theorem 3.7 then yields the following.

A recent result of Hinkkanen and Martin [Reference Hinkkanen and Martin9, Theorem 1] states that if p > 0, Ω is a plane domain and $\mathcal{F}$ is a family of K-quasiregular mappings $f:\Omega \to \mathbb{C}$ which is uniformly bounded in $L^p(\Omega)$, then $\mathcal{F}$ is finitely normal. Combining this with Corollary 3.10, we obtain the following.

It would be interesting to know if the same result holds in higher dimensions.

3.3. Quasiregular dynamics

If $f:S^n \to S^n$ and all its iterates are K-quasiregular for some $K\geq 1$, then f is called uniformly quasiregular. Just as in complex dynamics, the iteration theory of uniformly quasiregular mappings can be studied via normality. More precisely, the Fatou set F(f) is the set of stable behaviour where the family of iterates locally forms a normal family and the Julia set J(f) is the set of chaotic behaviour where the family of iterates does not locally form a normal family. The notion of normal family here is the usual version of Miniowitz with respect to Sⁿ, but as we have seen, this is equivalent to our notion of normality, compare with Definition 3.3.

For r > 0, let

\begin{equation*}L(x,f,r) = \sup_{\sigma (y,x)=r} \sigma ( f(y),f(x)).\end{equation*}

Applying the final statement of Theorem 3.7 leads to the following characterization of the Julia set of a uniformly quasiregular mapping.

Theorem 3.12. Let $f:S^n \to S^n$ be uniformly K-quasiregular. Then, $x_0\in J(f)$ if and only if there exist sequences $r_k \to 0$ and $m_k \to \infty$ such that

(3.2)\begin{equation} \lim_{k\to \infty} \frac{L(x_0,f^{m_k},r_k)}{r_k^{\alpha}} = \infty, \end{equation}

where $\alpha = K^{1/(1-n)}$.

Proof. Suppose $x_0 \in F(f)$. Then there exists $r_0 \gt 0$ such that the family $\{f^m |_{B_{\sigma}(x_0,r_0)} \}$ is normal. By the final statement of Theorem 3.7, this means that if $r\leq r_0/2$, there exists C > 0 such that

\begin{equation*} L(x_0,f^m,r) \leq Cr^{\alpha},\end{equation*}

for all $m \in \mathbb{N}$. It is then clear that (3.2) cannot hold for any sequences $r_k \to 0$ and $m_k \to \infty$. On the other hand, suppose $x_0 \in J(f)$ and let $r_k\to 0$. Fix $\delta \in (0,2)$. By [Reference Fletcher5, Lemma 4.2], for each k, we can find $m_k \in \mathbb{N}$ such that $L(x_0,f^{m_k},r_k) \geq \delta$. Since $r_k \to 0$, we obtain (3.2).

5.1. Bloch mappings

A Bloch function is a holomorphic function $f:\mathbb{D} \to \mathbb{C}$ satisfying

\begin{equation*} \sup _{z\in \mathbb{D} } (1-|z|^2)|f^{\prime}(z)| \lt \infty.\end{equation*}

Bloch functions have been much studied, see for example [Reference Pommerenke20]. Bloch functions can be characterized as holomorphic functions which are Lipschitz with respect to the hyperbolic distance in the domain and the Euclidean distance in the range, or equivalently, as functions where the family $\mathcal{F} = \{f(A(x)) - f(A(0)) : A\in G\}$ is normal, where G is the group of Möbius automorphisms of the disk, see [Reference Pommerenke18]. A profitable viewpoint for the study of Bloch functions has been to put a norm on the space of Bloch functions via

\begin{equation*} ||f|| _{\mathcal{B}} = \sup_{z\in \mathbb{D}} (1-|z|^2) |f^{\prime}(z)| + |f(0)|.\end{equation*}

The term $|f(0)|$ here is to be able to distinguish between constant functions, and without it, we just have a semi-norm.

In our quasiregular setting, a Bloch mapping is a special case of Definition 4.1(ii) when X is $\mathbb{B}^n$ equipped with the hyperbolic distance, and we take G to be the full conformal isometry group of $\mathbb{B}^n$. We will denote this group by G throughout this section.

It is evident from the discussion above that if f is a holomorphic function on $\mathbb{D}$, then it is a Bloch function if and only if it is a Bloch mapping in the sense of Definition 5.1. Bloch mappings exist in all dimensions as the following example shows.

We will give a characterization of Bloch mappings via the Bloch radius. If $f:\mathbb{B}^n \to \mathbb{R}^n$ is K-quasiregular, then following, for example, [Reference Eremenko4], we define the Bloch radius $\mathcal{B}_f$ to be the supremum of all radii r > 0 such that there exists a domain $U \subset \mathbb{B}^n$ with the property that $f|_U$ is bijective and f(U) is a ball of Euclidean radius r. We remark that Rajala [Reference Rajala21, Theorem 1.5] has proved a quantitative version of a result of Eremenko [Reference Eremenko4], namely that if f is as above, then

\begin{equation*} \mathcal{B}_f \geq C \operatorname{diam} f(B(0,1/2) ),\end{equation*}

where the diameter here is in the Euclidean distance, and C > 0 depends only on n and K.

This result is well-known for analytic functions in the disk. We thank the referee for pointing out that such a result appears in the work of Seidel and Walsh [Reference Seidel and Walsh25, Theorem 3], which pre-dates the terminology Bloch function.

Proof. First suppose that $\mathcal{B}_f$ is finite. Clearly, the Bloch radius is unchanged if we replace f by $f\circ A$ for $A\in G$. Then by [Reference Eremenko4, Theorem 1(iii)], the family $\{f\circ A : A\in G\}$ is equicontinuous with respect to the Euclidean distance in domain and range. In particular, if $E\subset \mathbb{B}^n$ is compact, then for any ϵ > 0, there exists δ > 0 so that if $x,y\in E$ with $|x-y| \lt \delta$, then $|f(A(x)) - f(A(y))| \lt \epsilon$. Since $\rho(x,y) \geq |x-y|$ and $\rho(A(x),A(y)) = \rho(x,y)$ for all $A\in G$, we conclude that f is uniformly continuous with respect to the hyperbolic and Euclidean distances. Theorem 4.2(ii) implies that f is normal and hence a Bloch mapping.

On the other hand, suppose that f is a Bloch mapping and the Bloch radius is infinite. Then there exist sequences $(x_m)$ in $\mathbb{B}^n$, $r_m\to \infty$ and neighbourhoods U_m of x_m so that f is bijective on U_m and $f(U_m) = B(f(x_m), r_m)$. Let $A_m \in G$ be a Möbius map sending 0 to x_m and define

\begin{equation*} g_m(x) = f(A_m(x)) - f(A_m(0)).\end{equation*}

Further, for $x\in \mathbb{B}^n$, set

\begin{equation*} h_m(x) = g_m^{-1}(r_m x),\end{equation*}

where we take the branch of the inverse corresponding to $(f|_{U_m})^{-1}$. Since each h_m is a K-quasiconformal map from $\mathbb{B}^n $ into $\mathbb{B}^n$, Theorem 3.5 implies $\{h_m : m \in \mathbb{N}\}$ is normal, and by passing to a subsequence if necessary, we may assume that $h_m \to h_0$ locally uniformly. Since $h_m(0)=0$ for all m, we conclude that $h_0(0)=0$.

Fix $x_0 \neq 0$ in $\mathbb{R}^n$ and, for m large enough so that $r_m \gt |x_0|$, find $w_m \in A_m^{-1}(U_m)$ so that $g_m(w_m) = x_0$. Moreover, let $u_m \in \mathbb{B}^n$ be such that $h_m(u_m) = w_m$. Since $u_m = x_0 / r_m$ by construction, we have $u_m \to 0$. Since $h_m\to h_0$ uniformly on compact subsets of $\mathbb{B}^n$ and $u_m\to 0$, we find that $h_m(u_m) \to h_0(0)=0$, that is, $w_m \to 0$.

We will now aim for a contradiction. To that end, choose $M\in\mathbb{N}$ large enough so that $w_m \in E= \overline{B_{\rho}(0,1)}$ for $m\geq M$. Since E is compact and $\{g_m : m\in \mathbb{N} \}$ is a finitely normal family of K-quasiregular maps, Corollary 3.10 implies the existence of a constant C > 0 so that

\begin{equation*} |g_m(x) | \leq C \rho (x,0)^{\alpha},\end{equation*}

for all $x\in E$. Applying this to w_m, and recalling that $g_m(w_m) = x_0$, we obtain

\begin{equation*} \rho(w_m,0) \geq \left ( \frac{|x_0|}{C} \right )^{1/\alpha},\end{equation*}

for $m\geq M$. This contradicts the earlier established fact that $w_m\to 0$.

An analogue for the Bloch norm in the quasiregular setting is the following.

Definition 5.5. Suppose $f:\mathbb{B}^n \to \mathbb{R}^n$ is a quasiregular Bloch mapping. Then we define

\begin{equation*} R_f := |f(0)| + \sup_{x\in \mathbb{B}^n} \operatorname{diam} f(B_{\rho}(x,1) ).\end{equation*}

Clearly, we cannot use the version of the Bloch norm for holomorphic functions in $\mathbb{D}$ since quasiregular mappings are only differentiable almost everywhere. Therefore, we can look at the largest Euclidean diameter of images of hyperbolic balls of a fixed radius instead. The choice of radius 1 here is arbitrary. In fact, if $0 \lt s \lt t$, then any hyperbolic ball in $\mathbb{B}^n$ of radius t can be covered by C balls of radius s, where C depends only on $n, s$ and t. In particular, whether R_f is finite or infinite is not changed if 1 is replaced by any other radius.

Proof. Suppose f is a Bloch mapping. Then since $E = \overline{B_{\rho}(0,1)}$ is compact, by Corollary 3.10, there exists C > 0 such that

(5.1)\begin{equation} |f(A(x)) - f(A(y)) | \leq C \rho (x,y)^{\alpha}, \end{equation}

for all $x,y \in E$ and all $A\in G$. Now, if $x_0 \in \mathbb{B}^n$, let $A_0 \in G$ map 0 to x ₀. Suppose $u \in B_{\rho}(x_0,1)$ and set $x=A_0^{-1}(u) \in E$. Then applying (5.1) with y = 0 and $A=A_0$, and using the fact that A ₀ is an isometry with respect to ρ, we see that

\begin{align*} |f(u)-f(x_0)| &= |f(A_0(x)) - f(A_0(0)) | \\ & \leq C \rho(x , 0)^{\alpha} \\ &= C\rho( A_0^{-1}(u) , A_0^{-1}(x_0) ) \\ &= C \rho(u, x_0)^{\alpha} \\ &\leq C, \end{align*}

for all $u \in B_{\rho}(x_0,1)$. Since x ₀ was arbitrary, it follows that $R_f \lt \infty$.

For the converse, suppose f is not a Bloch mapping and so by Theorem 5.4, the Bloch radius is infinite. As in the second half of the proof of Theorem 5.4, this means we can find sequences $(x_m)$ in $\mathbb{B}^n$, $r_m\to \infty$ and neighbourhoods U_m of x_m so that f is bijective on U_m and $f(U_m) = B(f(x_m), r_m)$. Moreover, let A_m be a Möbius mapping sending 0 to x_m, let $g_m(x) = f(A_m(x)) - f(A_m(0))$ and, for m large enough, let $h_m:\mathbb{B}^n\to\mathbb{B}^n$ be defined by $h_m = g_m^{-1}(r_mx)$.

Let $0 \lt r \lt 1$ and set $E = \overline{B(0,r)}$. Theorem 3.5 implies $\{h_m : m \in \mathbb{N}\}$ is finitely normal. Corollary 3.10 and Proposition 2.6 imply that $\{h_m :m\in \mathbb{N}\}$ is uniformly ω-continuous on E with respect to Euclidean distance. In particular, since $h_m(0) = 0$ for all m, there exists L > 0 so that

\begin{equation*} |h_m(x)| \leq L |x|^{\alpha}, \end{equation*}

for all $x\in E$. Find $0 \lt s\leq r$ so that $Ls^{\alpha } \lt ({(e-1)}/{(e+1)})$. Then

(5.2)\begin{equation} h_m(\overline{B(0,s)}) \subset \overline{B(0,Ls^{\alpha})} \subset \overline{B_{\rho}(0,1)}, \end{equation}

for all m. This implies that

\begin{equation*} g_m( \overline{B_{\rho}(0,1 ) } ) \supset \overline{B(0,r_ms)} .\end{equation*}

Since $r_m\to \infty$, we conclude that

\begin{equation*} \sup_{x\in \mathbb{B}^n} \operatorname{diam} f( B_{\rho}(x,1 )) = \infty,\end{equation*}

and hence $R_f = \infty$.

Let us remark that unlike the usual Bloch space of holomorphic Bloch functions, the set of quasiregular Bloch mappings does not form a normed vector space as it is not closed under addition. On the other hand, it is easy to see that $R_f \geq 0$ with equality if and only if f is the constant map 0.

Our next result shows that we can use R_f to quantify the growth of Bloch mappings.

Theorem 5.7. Let $f:\mathbb{B}^n \to \mathbb{R}^n$ be a quasiregular Bloch mapping. Then,

\begin{equation*} M(r,f) \leq R_f \max \left \{1 , \log \frac{1+r}{1-r} \right \}.\end{equation*}

Proof. Let r > 0 and suppose $|x_r| = r$ with $M(r,f) = |f(x_r)|$. Consider a chain of closed hyperbolic balls $U_1,\ldots, U_m$ each of hyperbolic radius 1, so that $U_{i} \cap U_{i+1}$ consists of one point, and with $0 \in \partial U_1$ and $x_r \in U_m$. This can be achieved with $m \leq \max \{1 , \rho(0,r) \}$ along a radial line segment. If $C = \sup_x \operatorname{diam} f(B_{\rho}(x,1) )$, then

\begin{equation*} |f(x_r)| \leq |f(0)| + Cm.\end{equation*}

Since $\rho(0,r) = \log {((1+r)}/{(1-r))}$, we conclude that

\begin{equation*} M(r,f) \leq |f(0)| + C\max \left \{1 , \log \left ( \frac{1+r}{1-r} \right ) \right \}\leq R_f \max \left \{1 , \log \frac{1+r}{1-r} \right \},\end{equation*}

as required.

Recalling that the order of growth of a map $f:\mathbb{B}^n \to \mathbb{R}^n$ is given by

\begin{equation*} \mu_f = \limsup_{r\to 1} (n-1) \frac{\log \,\log M(r,f)}{\log 1/(1-r)},\end{equation*}

we see that quasiregular Bloch mappings have zero order of growth.

In the literature, the little Bloch space $\mathcal{B}_0$ has been studied for holomorphic functions. There is an analogous definition in this setting.

Definition 5.8. We say that a quasiregular Bloch mapping is a little Bloch mapping if

\begin{equation*} \lim_{|x| \to 1} \operatorname{diam} f(B_{\rho}(x,1) ) = 0.\end{equation*}

For example, linear maps on $\mathbb{B}^n$ are clearly little Bloch mappings. There do exist Bloch mappings which are not little Bloch mappings, as is shown by the following natural generalization of the well-known example of the holomorphic Bloch map $\log(1-z)$.

Example 5.9. For simplicity, take n = 3 and consider a standard Zorich map construction $\mathcal{Z}$ such as that in [Reference Fletcher and Nicks6, Section 3.1] from the beam $\Omega = \{(a,b,c) : |a| \lt \pi/2, |b| \lt \pi/2 \}$ onto the upper half space $\{(a,b,c) : c \gt 0 \}$. The key point is that $\mathcal{Z}$ maps slices of the beam at height c = r onto hemispheres in the upper half space of radius e^r centred at the origin. With the given domain and range, $\mathcal{Z}$ is quasiconformal. Define $f:\mathbb{B}^3 \to \mathbb{R}^3$ via the map $f(x)= \mathcal{Z}^{-1}(x+(0,0,1))$.

Since $f(\mathbb{B}^3) \subset \Omega$, it is clear that the Bloch radius $\mathcal{B}_f$ of f is finite. By Theorem 5.4, we see that f is indeed a Bloch mapping.

We show that f is not a little Bloch mapping as follows. For $0 \lt r \lt 1$, take

\begin{equation*} x=(0,0,-r) \quad \mbox{and } \quad y=\left(0,0, -\frac{3r+1}{r+3}\right). \end{equation*}

Then, $\rho(x,y) = \log 2$, so $y\in B_\rho(x,1)$. Since the distance from the origin of $x+(0,0,1)$ is $1-r$, the third component of f(x) is $\log(1-r)$. Similarly, $|y+(0,0,1)|= {((2-2r)}/{(r+3))}$ and so the third component of f(y) is $\log{((2-2r})/{r+3}))$. Hence,

\begin{equation*} |f(x) - f(y)| \ge \left|\log(1-r) - \log\frac{2-2r}{r+3}\right| = \log\frac{r+3}{2}. \end{equation*}

Therefore, $\liminf_{r\to 1} \operatorname{diam} f(B_{\rho} (x,1) ) \ge\log 2 \gt 0$, and thus f is not a little Bloch mapping.

As a final remark in this section, holomorphic Bloch functions are studied with respect to their boundary behaviour. It would be interesting to study quasiregular Bloch mappings from this viewpoint too. We refer to [Reference Li and Villamor12] for the boundary behaviour of bounded quasiregular mappings in $\mathbb{B}^n$. It would be natural to ask whether the results there can be generalized to Bloch mappings.

5.2. Normal quasiregular mappings $\mathbb{B}^n \to S^n$

In this section, we consider normal quasiregular mappings $f:\mathbb{B}^n \to S^n$. This class has already been the subject of study, sometimes under the name normal quasimeromorphic mappings, and we refer to [Reference Vuorinen26, Chapter 13]. There, a quasiregular map $f:\mathbb{B}^n \to S^n$ is called normal if it is uniformly continuous with respect to the hyperbolic and spherical distances in the domain and range, respectively. By Theorem 4.2, this definition of a normal mapping agrees with Definition 4.1(i) with $X = \mathbb{B}^n$ equipped with the hyperbolic distance and G the full group of conformal isometries on $\mathbb{B}^n$.

Here, we prove a result on the order of growth of a normal quasiregular mapping. We refer to the terminology established in $\S$ 3.1, see also [Reference Rickman23, Chapter IV]. Given $f:\mathbb{B}^n \to S^n$ quasiregular, we modify (3.1) so that $A_{f,\sigma}^{\rho}(x_0,r)$ is the normalized volume of the image of a hyperbolic ball $f(\overline{B_{\rho}(x_0,r)})$ with multiplicity taken into account (see [Reference Rickman23, p.80]), that is,

\begin{equation*} A_{f,\sigma}^{\rho}(x_0,r) = \frac{1}{\omega_n} \int_{\overline{B_{\rho}(x_0,r)}} \frac{J_f(x)}{(1+|f(x)|^2)^n} \: {\rm d}m(x).\end{equation*}

Proof. Since $f:\mathbb{B}^n \to S^n$ is normal, f is uniformly continuous with respect to the hyperbolic and spherical distances, respectively, by Theorem 4.2. Hence, there exists δ > 0 such that

\begin{equation*} \sup_{x\in \mathbb{B}^n} \operatorname{diam}_{\sigma} f(B_{\rho}(x,2\delta) ) \lt \frac{1}{2}.\end{equation*}

Now let $x\in \mathbb{B}^n$ and find $M\in G$ such that $M(0)=x$. Let r > 0 and θ > 1 be chosen so that $B_{\rho}(0,\delta) = B(0,r)$ and $B_{\rho}(0,2\delta) = B(0,\theta r)$. Then

\begin{equation*} \operatorname{diam}_{\sigma} (f\circ M)(B(0,\theta r)) = \operatorname{diam} _{\sigma} f(B_{\rho}(x,2\delta)) \lt \frac{1}{2}.\end{equation*}

Pick an $(n-1)$-sphere of spherical radius $1/2$ such that $Y \cap (f\circ M)(B(0,\theta r)) = \emptyset$. Then $\nu_{f\circ M} (\theta r, Y) = 0$. By [Reference Rickman23, Lemma IV.1.7] applied to $f\circ M$, we have

\begin{equation*} A_{f,\sigma}^{\rho}(x,\delta) = A_{f\circ M,\sigma}^{\rho}(0,\delta) = A_{f\circ M, \sigma}( r ) \leq \frac{K_I Q (\log 2)^{n-1} }{(\log \theta )^{n-1} },\end{equation*}

where Q > 0 depends only on n, and we recall that K_I is the inner dilatation of f. Since this holds for all $x\in \mathbb{B}^n$, the lemma follows.

Proof. Let $r\in (0,1)$ and $t = \rho(0,r)$. Then $A_{f,\sigma}^{\rho}(0,t) = A_{f,\sigma}(0,r)$. By [Reference Ratcliffe22, p.79], the hyperbolic volume of $B_{\rho}(x,t)$ is

\begin{equation*} \operatorname{Vol}_{\rho}(B_{\rho}(x,t)) = \omega_{n-1} \int_0^t \sinh ^{n-1} u \,{\rm d}u \sim C_n \,{\rm e}^{(n-1)t}\end{equation*}

as $t\to \infty$, where C_n depends only on n. Here $f(t) \sim g(t)$ as $t\to \infty$ means that $f(t)/g(t) \to 1$ as $t\to \infty$.

Hence, given δ > 0, the number of balls of hyperbolic radius δ needed to cover $B_{\rho}(x,t)$ is bounded above by $D_n\,{\rm e}^{(n-1)t}$ for large t, where D_n depends only on n and δ. Then, by Lemma 5.11, and using the constants δ and C from Lemma 5.11, we have

\begin{equation*} A_{f,\sigma}(0,r) = A_{f,\sigma}^{\rho}(0,t) \leq CD_n\,{\rm e}^{(n-1)t} = CD_n \left ( \frac{1+r}{1-r} \right )^{n-1}.\end{equation*}

Hence,

\begin{equation*} \mu_f = \limsup_{r\to 1} \frac{\log A_{f,\sigma}(0,r) }{\log \frac{1}{1-r} } \leq n-1.\end{equation*}

We observe that [Reference Vuorinen26, Theorem 13.10] gives an order of growth estimate depending on n and K (the constant γ in that result) related to normal quasiregular mappings $\mathbb{B}^n\to S^n$ without poles. Theorem 5.12 can be viewed as a variation of this result with the dependence on K dropped and poles of f permitted.

5.3. Yosida mappings

A Yosida function in the plane is a meromorphic function $f:\mathbb{C} \to S^2$ such that the family $\mathcal{F} = \{f(z+w) : w\in \mathbb{C} \}$ is normal. We make an analogous definition by using Definition 4.1(i) with $X = \mathbb{R}^n$ equipped with the Euclidean distance, and G the translation group of $\mathbb{R}^n$.

Using an approach similar to that behind Theorem 5.12, we can recover a bound on the order of growth of Yosida mappings contained in [Reference Bonk and Heinonen2, Corollary 2.2].

Proof. Since f is a Yosida mapping, Theorem 4.2 implies that f is uniformly continuous with respect to the Euclidean and spherical distances. Hence, there exists δ > 0 so that

\begin{equation*} \sup _{x\in \mathbb{R}^n} \operatorname{diam} _{\sigma} f(B(x,2\delta)) \lt \frac12.\end{equation*}

The same argument as in Lemma 5.11 (except without needing to amend $A_{f,\sigma}$ to include the hyperbolic distance) implies the existence of C > 0 such that

\begin{equation*} A_{f,\sigma}(x,\delta) \leq C, \end{equation*}

for all $x\in \mathbb{R}^n$. Since there exists a constant $D = D(n,\delta)$ so that we may cover a ball of large radius r by at most $D r^n$ balls of radius δ, we have

\begin{equation*} A_{f,\sigma} (0,r) \leq CDr^n.\end{equation*}

Hence,

\begin{equation*} \mu_f = \limsup_{r\to \infty} \frac{\log A_{f,\sigma}(0,r) }{\log r} \leq n. \end{equation*}

If $f:\mathbb{C} \to \mathbb{C}$ is holomorphic and also a Yosida function, then the order is at most 1, see [Reference Clunie and Hayman3, Theorem 3]. We believe the analogous result is true for entire quasiregular mappings, and hence we make the following conjecture.

Examples of Yosida mappings include Zorich mappings and certain other periodic quasiregular mappings, see for example [Reference Martio and Srebro14, Sections 7 and 8]. The standard Zorich mappings on $\mathbb{R}^n$ have order n − 1 (and more generally see [Reference Martio and Srebro14, Corollary 8.11]), which would imply the sharpness of the above conjecture.

Note that Theorem 4.4 states in this setting that a K-quasiregular mapping $f:\mathbb{R}^n \to S^n$ is Yosida if and only if it is α-Hölder continuous, that is, there exists C > 0 so that

(5.3)\begin{equation} \sigma(f(x),f(y)) \leq C|x-y|^{\alpha}, \end{equation}

for all $x,y\in\mathbb{R}^n$, recalling that $\alpha = K^{1/(1-n)}$. We will prove the following characterization of Yosida mappings, generalizing [Reference Minda15, Theorem 1] for Yosida functions in the plane and closely following the proof of Miniowitz’s version of Zalcman’s Lemma [Reference Miniowitz16, Lemma 1].

Proof. First, suppose that there are sequences $x_m,\rho_m$ as in the hypotheses and a non-constant Yosida mapping g so that $f(x_m + \rho_mw) \to g(w)$ locally uniformly on $\mathbb{R}^n$. Further, assume that f is a Yosida mapping. Then by (5.3), for any $w\in \mathbb{R}^n$, we have

\begin{equation*} \operatorname{diam}_{\sigma}( g( \overline{B(w,1) } ) ) = \lim_{m\to \infty} \operatorname{diam}_{\sigma} ( f ( \overline{B(x_m + \rho_mw, \rho_m ) } ) ) \leq \lim_{m\to \infty} C (2 \rho_m)^{\alpha}.\end{equation*}

It follows that $\operatorname{diam}_{\sigma} ( g( \overline{B(w,1) } ) ) = 0$. This implies that g is constant, yielding a contradiction.

For the converse, suppose now that $f:\mathbb{R}^n \to S^n$ is not a Yosida mapping and so ${\mathcal{F} = \{f(x+a) : a\in \mathbb{R}^n \}}$ is not a normal family. Then, $\mathcal{F}$ is not normal on $B(0,r^*)$ for some $r^* \gt 0$. By Theorem 3.7, there exists $f_m \in \mathcal{F}$ and $x_m^*,y_m^* \in B(0,r^*)$ with

\begin{equation*} \lim_{m\to \infty} \frac{\sigma (f_m(x_m^*) , f_m(y_m^*) ) }{|x_m^* - y_m^* |^{\alpha} } = \infty.\end{equation*}

Fix $r \gt r^*$ and set

\begin{equation*} M_m = \sup_{|x|,|y| \leq r } \left ( 1 - \frac{|x|^2}{r^2} \right) \frac{\sigma( f_m(x) , f_m(y) )}{|x-y|^{\alpha} }.\end{equation*}

Clearly $M_m \to \infty$. Now choose $x_m,y_m \in B(0,r)$ with

\begin{equation*} \left ( 1 - \frac{|x_m|^2}{r^2} \right ) \frac{\sigma (f_m(x_m) , f_m(y_m) ) }{|x_m - y_m|^{\alpha} } \geq \frac{M_m}{2}\end{equation*}

and set

(5.4)\begin{equation} g_m(w) = f_m( x_m + \rho_mw), \quad \mbox{where } \ \rho_m^{\alpha} = \frac{|x_m - y_m| ^{\alpha} }{\sigma ( f_m(x_m) , f_m(y_m) )}. \end{equation}

Since

\begin{equation*} \rho_m^{\alpha} \leq \left ( \frac{2}{M_m} \right ) \left ( 1 - \frac{|x_m|^2}{r^2} \right ),\end{equation*}

we have $\rho_m \to 0$ as $m \to \infty$. Now, $|x_m + \rho_m w| \leq r$ if $|w|\leq R_m$, where $R_m = (r-|x_m|)/\rho_m$. As

\begin{equation*} \frac{1}{R_m} = \frac{\rho_m}{r - |x_m| } \leq \frac{2(r+|x_m| )\rho_m^{1-\alpha} }{r^2 M_m} \leq \frac{4\rho_m^{1-\alpha} }{rM_m},\end{equation*}

and α < 1, it follows that $R_m \to \infty$ as $m\to \infty$. We next show that $(g_m)$ forms a normal family on $\mathbb{R}^n$.

Let R > 0 and let $w_1,w_2 \in B(0,R)$. Choose m large enough that $R \lt R_m$ and so $|x_m + \rho_m w_i | \lt r$ for $i=1,2$. Then by the definitions of g_m and M_m, we have

(5.5)\begin{align} \frac{\sigma (g_m(w_1) , g_m(w_2) ) }{|w_1 - w_2|^{\alpha}} &= \frac{\sigma ( f_m( x_m + \rho_m w_1 ) , f_m( x_m + \rho_m w_2) )}{|w_1 - w_2 |^{\alpha} } \\ &= \frac{\rho_m^{\alpha} \sigma ( f_m ( x_m + \rho_m w_1) , f_m ( x_m + \rho_m w_2 ) ) }{| ( x_m + \rho_m w_1) - (x_m + \rho_mw_2) |^{\alpha} } \\ & \leq \frac{\rho_m^{\alpha} M_m}{ 1 - \frac{|x_m + \rho_m w_1 | ^2}{r^2} } \\ & \leq \frac{\left ( \frac{2}{M_m} \right ) \left ( 1 - \frac{|x_m|^2}{r^2} \right ) M_m}{1 - \frac{|x_m + \rho_mw_1|^2}{r^2} } \\ &= \frac{2 ( r^2 - |x_m|^2) }{r^2 - |x_m + \rho_mw_1|^2} \\ & \leq 2 \left ( \frac{r+|x_m|}{r+|x_m| +R\rho_m } \right ) \left ( \frac{r-|x_m|}{r-|x_m| - R\rho_m } \right ). \end{align}

The first ratio here is bounded above by 1 and the second converges to 1 as $m\to \infty$ for R fixed. Consequently, by Theorem 3.7, $(g_m)$ forms a normal family.

We need to show that a limit g of a convergent subsequence of $(g_m)$ is non-constant. To that end, find ξ_m with $x_m+\rho_m\xi_m = y_m$. Then, by (5.4)

\begin{equation*} |\xi_m| = \frac{|y_m - x_m|}{\rho_m} = \frac{\left|y_m - x_m\right| \sigma( f_m(x_m) , f_m(y_m) )^{1/\alpha} }{|x_m-y_m|} \leq \pi^{1/\alpha}.\end{equation*}

Therefore, $(\xi_m)$ contains a convergent subsequence which we pass to and relabel. Note that

(5.6)\begin{equation} \sigma(g_m(0) , g_m(\xi_m) ) = \sigma(f_m(x_m) , f_m(y_m) ) = \frac{|x_m-y_m|^{\alpha}}{\rho_m^{\alpha} } = |\xi_m|^{\alpha}. \end{equation}

We may assume that $\xi_m\to0$, since otherwise any limit function g will take distinct values at 0 and $\lim \xi_m$. Now let $h_m = U_m \circ g_m$, where U_m is a spherical isometry sending $g_m(0)$ to 0. By the equicontinuity of $(g_m)$, we can find a > 0 such that the h_m are uniformly bounded K-quasiregular mappings on $B(0,a)$ for all large m and hence form a normal family. We have

(5.7)\begin{equation} \sigma(g_m(0) , g_m(\xi_m) ) = \sigma(h_m(0) , h_m(\xi_m) ) \le 2 |h_m(\xi_m)|, \end{equation}

recalling the fact that $\sigma(x,y)\le 2|x-y|$. For large m, we now apply Theorem 3.1 with ${U = B(0,a)}$, ${f= h_m}$, ${E = \{0 \}}$, ${y = \xi_m \in B(0,a)}$, ${\delta = d(0,\partial U) = a}$ and ${r= \operatorname{diam}(h_m(B(0,a)))}$ to get

\begin{equation*} |h_m(\xi_m)| \leq \frac{\lambda(n) |\xi_m|^{\alpha} \operatorname{diam}(h_m(B(0,a))) }{a^{\alpha} } .\end{equation*}

Recall that $\lambda(n)$ is a constant depending only on n. Combining this with (5.6) and (5.7), we conclude that

\begin{equation*} \operatorname{diam}(h_m(B(0,a))) \geq \frac{a^{\alpha}}{2\lambda(n)}.\end{equation*}

Consequently, once we pass to a convergent subsequence of $(h_m)$, we see that any limit must be non-constant. Since $h_m = U_m \circ g_m$, the same is therefore true for any limit of a subsequence of $(g_m)$.

Finally, we have to show that any such limit g is a Yosida mapping. For any R > 0 and any $w_1,w_2 \in B(0,R)$, (5.5) implies that

\begin{equation*} \sigma ( g(w_1) , g(w_2) ) \leq 2|w_1-w_2|^{\alpha}.\end{equation*}

Consequently, this is also true for any $w_1,w_2 \in \mathbb{R}^n$. This shows that $g:\mathbb{R}^n \to S^n$ is globally Hölder continuous, and hence by Theorem 4.4, g is normal. In other words, g is a Yosida mapping, as required.