Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 24, 1999, 231–252

NORMAL FAMILIES, MULTIPLICITY AND THE BRANCH SET OF QUASIREGULAR MAPS

O. Martio, U. Srebro, and J. Väisälä

University of Helsinki, Department of Mathematics P.O. Box 4, FIN-00014 Helsinki, Finland; olli.martio@helsinki.fi Technion, Department of Mathematics Haifa 32000, Israel; srebro@math.technion.ac.il University of Helsinki, Department of Mathematics P.O. Box 4, FIN-00014 Helsinki, Finland; jussi.vaisala@helsinki.fi

Abstract. A criterion for normality and compactness of families of K -quasiregular mappings of bounded multiplicity is established and then applied to the study of the branch set and its image.

1. Introduction

Let D be a domain in R^{n}, n ≥ 2, and let f: D → R^{n} be a discrete and open map. By a theorem of Chernavskii [C1]–[C2], see also [V1], both the branch set B_{f} of f, i.e. the set where f fails to define a local homeomorphism, and fB_{f} have topological dimension ≤ n − 2. For n = 2, B_{f} consists of isolated points, the local behavior of f at a point x ∈ D is quite simple, and it is classified by its local topological index i(x, f). Contrary to the planar case, little is known of the structure of B_{f} for n ≥ 3, and maps with the same index i(x, f) at x may have different topological behavior in any neighborhood of x. Even for n = 3 and for small values of i(x, f), the local behavior of a discrete open map can be complicated unless the image of the branch set is relatively simple near the point f(x); see [MRV3, 3.20] and [MSr, 3.8].

Suppose that f: D → R^{n}, n ≥ 2, is quasiregular. This means that f is continuous, locally in the Sobolev space W^{1,n}, and for some K ≥ 1

(1.1)  |f′(x)|^{n} ≤ KJ(x, f)

a.e. in D. Here f′(x) is the formal derivative of f at x, |f′(x)| = sup_{|h|=1} |f′(x)h| and J(x, f) = det f′(x) is the Jacobian determinant of f at x. By a theorem of Reshetnyak [Re, pp. 183–184], f is either constant or a discrete, open and sensepreserving map. We shall only consider the latter case. Since quasiregular maps form a natural generalization of plane analytic functions to higher dimensional euclidean spaces, rather many studies have been devoted to the metric structure of their branch sets.

1991 Mathematics Subject Classification: Primary 30C65.

The research of the second author was partially supported by grants from the Finnish Academy, the Fund of Promotion of Research at the Technion and Japan Technion Society Research Fund.

In [MRV1] it was shown that m(B_{f}) = 0 = m(fB_{f}), where m refers to the Lebesgue measure in R^{n}; see also [Re, p. 224]. Moreover, on each (n − 1)- hyperplane T, m_{n−1}(T ∩ B_{f}) = 0 = m_{n−1}(T ∩ fB_{f}); see [Re, p. 221] and [MR, 3.1] for these results. Sarvas [S2, 4.10] showed that for any compact set C ⊂ D, dim_{ℋ} (B_{f} ∩ C) < n where dim_{ℋ} refers to the Hausdorff dimension. In [MRV3, 4.4] it was proved that if n ≥ 3 and if B_{f} omits an open cone C_{x}(α) with vertex at x and opening angle α > 0, then i(x, f) ≤ N(α, K, n). We replace the cone C_{x}(α) by a curvilinear cone and show in Section 5 that this result is quantitatively the best possible.

In this paper we study metric properties of the domain D\B_{f}, assuming that f: D → R^{n} is a K -quasiregular map of finite multiplicity

N(f) = sup {#f^{−1}(y) : y ∈ R^{n}} < ∞.

For example, we show in Section 3 that if D = R^{n}, then R^{n} \ B_{f} and R^{n} \ fB_{f} are uniform domains, and hence contain arbitrarily large balls. In fact, there are arbitrarily large balls in which f is injective. The proofs are based on normal family properties of quasiregular maps of finite multiplicity. These are studied in Section 2, and they differ considerably from the quasiconformal case. In Section 4 we show that the class of nullsets for uniform domains and the class of porous sets are invariant under quasiregular maps f: R^{n} → R^{n} of finite multiplicity. The results hold in all dimensions n ≥ 2.

Iwaniec [Iw] has studied normal families and injectivity of quasiregular mappings. His studies were mainly devoted to the stability problem, i.e. to quasiregular mappings in R^{n}, n ≥ 3, whose dilatation coefficient K is close to 1. He also uses a different type of normalization.

Our notation is standard. In particular, B^{n}(x, r) or B(x, r) denotes the open ball centered at x ∈ R^{n} with radius r > 0, B^{n}(r) = B^{n}(0, r) and B^{n} = B^{n}(1). Also S^{n−1}(x, r) = ∂B^{n}(x, r), S^{n−1}(r) = S^{n−1}(0, r) and S^{n−1} = S^{n−1}(1). For A ⊂ R^{n} and r > 0, we let B(A, r) = {x : dist (x, A) < r} denote the rneighborhood of A with B(A, ∞) = R^{n}. The one-point extension of R^{n} is Ṙ^{n} = R^{n} ∪ {∞}. For real numbers r, s we write r ∧ s = min (r, s).

2. Normalization and normality

Normal families, multiplicity and the branch set of quasiregular maps 233

Let D be a domain in R^{n}, n ≥ 2, let 1 ≤ N < ∞, and let ℱ denote the family of all K -quasiregular maps f: D → R^{n} with N(f) ≤ N. Clearly, ℱ is invariant under the action of sense preserving similarities of R^{n}, i.e. A ∘ f ∈ ℱ if f ∈ ℱ and A is a sense preserving similarity. Next, let φ: ℱ → R be a functional, and suppose that φ is invariant under sense preserving similarities, i.e. φ(A ∘ f) = φ(f), where A and f are as above.

In studying the infimum of φ on ℱ one often considers a sequence (f_{k}) of elements of ℱ such that

lim_{k→∞} φ(f_{k}) = inf_{f∈ℱ} φ(f).

In view of the similarity invariance of ℱ and φ, one may replace each f_{k} by another element g_{k} ∈ ℱ which satisfies certain normalization conditions, such as

(2.1)  g_{k}(a) = a  and  g_{k}(b) = b

for two fixed points a and b in D.

In the case where n = 2, K = 1 and N = 1, the maps are complex analytic univalent functions, and one can normalize the maps g_{k} also by the condition

(2.2)  g_{k}(a) = a  and  g′_{k}(a) = 1.

In this case, each of the conditions (2.1) and (2.2) implies normality, and this fact is widely used in the theory of analytic univalent functions, cf. [Po2] and [Sc]. This, however, is not the case as soon as N > 1, as noted already in [Po2] and can be seen from the following two examples. The functions

g_{k}(z) = (k + 1)z − kz^{2}, z ∈ C,

k = 1, 2, …, are analytic and 2-valent in B^{2}(r) for any r > 1. They satisfy (2.1) with a = 0 and b = 1, but (g_{k}) is not normal in any neighborhood of 0 since g_{k}(1/k) = 1, k = 1, 2, …. The functions

g_{k}(z) = z − kz^{2}, z ∈ C,

k = 1, 2, …, are analytic and 2-valent in B^{2}(r) for any r > \frac{1}{2} . They satisfy (2.2) with a = 0, but (g_{k}) is not normal in any neighborhood of 0 because g_{k}(k^{−1/3}) → ∞ as k → ∞. These two examples can be generalized to quasiregular maps in all dimensions n ≥ 2 showing that another normalization is needed for noninjective maps.

Let ℱ and φ be as above. Choose a point a ∈ D and a number R > 0 such that \overline{B}(a, R) ⊂ D. Then, by the similarity invariance of ℱ and φ, for each f ∈ ℱ there exists g ∈ ℱ with φ(g) = φ(f) such that

g(a) = 0  and  max_{|x−a|=R} |g(x)| = 1.

We show in 2.5 that this normalization yields a normal, and even compact, family of elements g of ℱ in B(a, R). This will follow from a more general result 2.4, which will be needed in Section 3. We recall that a family ℱ of maps f: D → R^{n} is normal if from each sequence of functions f_{k} ∈ ℱ it is possible to extract a subsequence (f_{k_{i}}) which converges locally uniformly in D to a function f: D → Ṙ^{n}.

We first prove a distortion lemma.

2.3. Lemma. Let 0 < r < s < R ≤ ∞, 1 ≤ K < ∞, N ≥ 1 and n ≥ 2. Then there is c = c(r, s, R, K, N, n) with the following property: If f: B^{n}(R) → R^{n} is a K -quasiregular map with f(0) = 0 and N(f) ≤ N, and if A ⊂\overline{B}^{n}(r) is a continuum joining 0 and S^{n−1}(r), then

max{|f(x)| : |x| ≤ s} ≤ c max{|f(x)| ∈ A}.

Proof. Let m_{0} and m_{1} be the maximum of |f(x)| over x ∈ A and |x| ≤ s, respectively. Choose x ∈ S^{n−1}(s) with |f(x)| = m_{1} , and define a path α: [1, ∞) → R^{n} by α(t) = tf(x). Let α^{∗}: [1, t_{0}) → B^{n}(R) be a maximal lift of α starting at x; see [MRV3, 3.11]. Then |α^{∗}(t)| ≥ s for all t, and |α^{∗}(t)| → R as t → t_{0} .

Let Γ be the family of all paths joining A and the locus of α^{∗} in B^{n}(R). For the modulus M(Γ), a standard estimate gives a lower bound

M(Γ) ≥ q(r, s, R, n) > 0;

see [GM, 2.6 and 2.12]. We may assume that m_{0} < m_{1} . Since each member of fΓ meets the spheres S^{n−1}(m_{0}) and S^{n−1}(m_{1}), we have

M(fΓ) ≤ ω_{n−1}(log \frac{m_{1}}{m_{0}} )^{1−n}.

Since f is K -quasiregular with N(f) ≤ N, the K_{O}(f)-modulus inequality [MRV1, 3.2] yields M(Γ) ≤ KNM(fΓ). Combining these inequalities we obtain the lemma.

2.4. Theorem. Suppose that 0 < r < R ≤ ∞, 0 < r′ < ∞, 1 ≤ K < ∞, N ≥ 1, and that ℱ is a family of K -quasiregular maps f: B^{n}(R) → R^{n} such that N(f) ≤ N, f(0) = 0, and such that for each f ∈ ℱ there is a continuum A(f) with the properties

0 ∈ A(f),  max{|x| : x ∈ A(f)} = r,  max{|f(x)| : x ∈ A(f)} = r′.

Then ℱ is a normal family. If f_{k} ∈ ℱ and if f_{k} → f locally uniformly in B^{n}(R), then f is a K -quasiregular map with N(f) ≤ N.

Proof. For r < s < R, Lemma 2.3 implies that |f(x)| ≤ c(r, s, R, K, N, n)r′ for all |x| ≤ s and f ∈ ℱ. Thus ℱ is uniformly bounded in B^{n}(s), and the normality of ℱ follows from [MRV2, 3.17] or from [Re, p. 220].

Next let (f_{k}) be a sequence in ℱ converging to a map f locally uniformly in B^{n}(R). By a theorem of Reshetnyak [Re, p. 218], f is K -quasiregular. For each k there is a point x_{k} ∈ A(f_{k}) ∩ S^{n−1}(R) with |f_{k}(x_{k})| = r′. Hence |f(x)| = r′ for some x ∈\overline{B}^{n}(r). Since f(0) = 0, f is nonconstant. The inequality N(f) ≤ N follows by an easy degree argument.

2.5. Corollary. Let 1 < R ≤ ∞, 1 ≤ K < ∞ and 1 ≤ N < ∞, and let ℱ be a family of K -quasiregular maps f: B^{n}(R) → R^{n} with N(f) ≤ N satisfying

f(0) = 0  and  max_{|x|=1} |f(x)| = 1.

Then ℱ is a normal family. Moreover, if f_{k} ∈ ℱ and f_{k} → f locally uniformly in B^{n}(R), then f is K -quasiregular and N(f) ≤ N.

2.6. Remark. The assumption in 2.5 that N(f) ≤ N for all f ∈ ℱ is indispensable as can be seen by considering z^{k}, k = 1, 2, …, for n = 2, and a sequence of polynomial-like K -quasiregular maps f: R^{n} → R^{n} in the sense of [Ri, I.3.2] or [Mr, Th. 2] for n > 2.

3. The branch set and multiplicity

3.1. Terminology. Let c ≥ 1. A set A ⊂ R^{n} is c-plump if for each x ∈ Ā and for each r > 0 with A \ B(x, r) ≠ ∅ there is z ∈\overline{B}(x, r) such that B(z, r/c) ⊂ A. A set F ⊂ R^{n} is c-porous if int F = ∅ and if R^{n} \ F is c-plump.

Let D be a proper subdomain of R^{n}. For each x ∈ D we write

δ(x) = δ_{D}(x) = dist (x, ∂D).

A domain D is c-uniform if D = R^{n} or if each pair of points a, b in D can be joined by a rectifiable path γ: [0, l(α)] → D, parametrized by arc length, such that l(γ) ≤ c|a − b| and such that

(3.2)  t ∧ (l(α) − t) ≤ cδ(γ(t))

for all t ∈ (0, l(α)); see [MS] and [V4]. Recall that t ∧ s denotes min(t, s).

We recall from [MRV1] some basic properties of a discrete open map f: D → R^{n}. A domain U is a normal domain of f if \overline{U} is compact in D and if f∂U = ∂fU. For x ∈ D, the x-component U(x, f, r) of f^{−1}B(f(x), r) is a normal domain of f whenever its closure is compact in D. Then fU(x, f, r) = B(f(x), r). If, in addition, U(x, f, r) meets f^{−1}(f(x)) only at x, it is called a normal neighborhood of x. If U is a normal domain of f, then f defines a proper map U → fU, that is, the preimage of every compact set is compact.

For each x ∈ D there is r_{0} > 0 such that U(x, f, r_{0}) is a normal neighborhood of x for each r ≤ r_{0} , and diam U(x, f, r) → 0 as r → 0. Moreover, the topological degree μ(f(x), f, U(x, f, r)) is independent of r ∈ (0, r_{0}], and it is the local index i(x, f) of f at x. We also have |i(x, f)| = N(f | V) for every neighborhood V ⊂ U(x, f, r_{0}) of x. A point x ∈ D is in B_{f} if and only if |i(x, f)| ≥ 2. Nonconstant quasiregular maps are sense-preserving, that is, i(x, f) > 0 for all x ∈ D.

If f: R^{n} → R^{n} is quasiregular with N(f) = N < ∞, then f extends to a continuous map f: Ṙ^{n} → Ṙ^{n} by f(∞) = ∞. This follows, for example, from [Ri, III.2.11]. Consequently, f is a proper map and also a closed map onto R^{n}.

As main results of this section we shall show that if f: D → R^{n} is K - quasiregular with N(f) ≤ N < ∞ and if D is c-plump or c-uniform, then D \ B_{f} has the same properties with a constant c′ = c′(c, N, K, n).

For plumpness this follows from the following stronger result.

3.3. Theorem. For each n ≥ 2, K ≥ 1 and N ≥ 1 there exists q = q(N, K, n) > 0 such that f is injective in some ball B(x, q) ⊂ B^{n} whenever f: B^{n} → R^{n} is K -quasiregular with N(f) ≤ N.

Proof. Assume that the theorem is false for some triple (N, K, n). Then there is a sequence of K -quasiregular maps f_{k}: B^{n} → R^{n} with N(f_{k}) ≤ N such that f_{k} is not injective in any ball B(x, 1/k) ⊂ B^{n}. By auxiliary similarities we can normalize the maps f_{k} so that

f_{k}(0) = 0, max\{|f_{k}(x)| : |x| ≤ \frac{1}{2}\} = 1.

Applying 2.5 and passing to a subsequence we may assume that the sequence (f_{k}) converges locally uniformly to a nonconstant K -quasiregular map f: B^{n} → R^{n}. Choose a ball \overline{B}(a, r) ⊂ B^{n} in which f is injective. Then the topological degree μ(f(x), f, B(a, r)) is 1 for all x ∈ B(a, r). Since f_{k} → f uniformly in \overline{B}(a, r), there is k_{0} such that μ(f_{k}(x), f_{k}, B(a, r)) = 1 for all k ≥ k_{0} and x ∈ B(a, r/2). Hence f_{k} | B(a, r/2) is injective for all k ≥ k_{0} , which gives a contradiction.

3.4. Theorem. Suppose that D ⊂ R^{n} is a c-plump domain and that f: D → R^{n} is a K -quasiregular map with N(f) ≤ N < ∞. Then D \ B_{f} is c′ -plump with c′ = c′(c, N, K, n).

Proof. Assume that x ∈ \overline{D}, r > 0, and D \ B(x, r) ≠ ∅. Since D is cplump, there is z ∈ \overline{B}(x, r) with B(z, r/c) ⊂ D. By 3.3, f is injective in some ball B(y, qr/c) ⊂ B(z, r/c) where q = q(N, K, n). Hence D \ B_{f} is (c/q)-plump.

3.5. Corollary. If f: R^{n} → R^{n} is K -quasiregular with N(f) ≤ N < ∞, then B_{f} is c-porous with c = c(N, K, n). In particular, R^{n} \ B_{f} contains arbitrarily large balls.

3.6. Remark. Every K -quasiconformal map f: R^{n} → R^{n} is η -quasisym-metric with η = η_{K,n}; see [V3, 2.5]. Hence f maps each c-porous set in R^{n} onto a c′ -porous set, c′ = c′(c, K, n). In particular, the image of each (n−2)-dimensional plane T ⊂ R^{n} is c′ -porous with c′ = c′(K, n). We remark that this result can also be obtained from 3.5. Indeed, there is a 2-valent quasiregular winding map w: R^{n} → R^{n} with B_{w} = T, and then fT is the branch set of the quasiregular map w ∘ f^{−1} .

To obtain the uniform version of 3.4 we need some auxiliary results.

3.7. Terminology. The relative distance between points a, b in a domain D ≠ R^{n} is the number

r_{D}(a, b) = \frac{|a − b|}{δ(a) ∧ δ(b)},

where δ(x) = dist (x, ∂D) as before. For c ≥ 1, we say that a pair (a, b) of points in D is a c-pair in D if 1 ≤ r_{D}(a, b) ≤ c. This is a simplified version of the notion considered in [V4, 2.13].

3.8. Lemma. Suppose that D ≠ R^{n} is a domain and that

(1) D is c-plump.

(2) For each 8c-pair (a, b) in D there is an arc γ joining a and b such that

δ(a) ∧ δ(b) ≤ c_{0} dist (γ, ∂D), diam γ ≤ c_{0}|a − b|.

Then D is a c′ -uniform domain with c′ = c′(c, c_{0}, n).

Proof. Suppose that B(a, r) and B(b, s) are balls in D such that r/s ∈ [1/2, 2] and |a − b| ≤ 4c max(r, s). By [V4, 2.15 and 2.10], it suffices to show that a and b can be joined by an arc γ such that diam γ ≤ c_{1}|a − b| and such that

|x − a| ∧ |x − b| ≤ c_{1}δ(x)

for all x ∈ γ with c_{1} = c_{1}(c, c_{0}).

If r_{D}(a, b) ≤ 1, we can choose γ to be the line segment [a, b]. If r_{D}(a, b) ≥ 1, then (a, b) is obviously an 8c-pair in D. Hence there is γ satisfying (2). For each x ∈ γ we have

|x − a| ∧ |x − b| ≤ diam γ ≤ c_{0}|a − b| ≤ 8cc_{0}(δ(a) ∧ δ(b))
≤ 8cc_{0}^{2}dist (γ, ∂D) ≤ 8cc_{0}^{2}δ(x),

and the lemma is proved.

3.9. Lemma. Suppose that a and b are points in a c-uniform domain D ⊂ R^{n} such that 0 < |a − b| ≤ c′(δ(a) ∧ δ(b)). Then there is L = L(c, c′) ≥ 1 and an L-bilipschitz map F: B^{n}(|a−b|) → D such that F (0) = a and F (|a−b|e_{1}/2) = b.

Proof. Set r = δ(a) ∧ δ(b) and t = |a − b|. Then 0 < t ≤ c′r. The assertion is clear if r < t, since then B((a + b)/2, t\sqrt{3} /2) ⊂ D. By a result of G. Martin [Ma, 5.1], there is L = L(c) and an L-bilipschitz map f:\overline{B}^{n}(t) → D such that \{a, b\} ⊂ f\overline{B}^{n}(t). Set U = fB^{n}(t). Since f is L-bilipschitz, U is easily seen to be 2L^{2} -plump. Hence there is a ball B(z, s) ⊂ U ∩ B(a, r/4) with s = r/16L^{2} . It follows that there is L_{1} = L_{1}(c) and an L_{1} -bilipschitz homeomorphism g:\overline{B}(a, r/2) → \overline{B}(a, r/2) such that g | ∂B(a, r/2) = id and such that gB(z, s) = B(a, s). Since r ≤ t, the balls B(a, r/2) and B(b, r/2) are disjoint. Hence we can use the same construction in B(b, r/2) to extend g to an L_{1} -bilipschitz homeomorphism g: R^{n} → R^{n} such that

(1) B(a, s) ∪ B(b, s) ⊂ gU,

(2) g = id outside B(a, r/2) ∪ B(b, r/2).

Setting h(x) = g(f(x)) we obtain an LL_{1} -bilipschitz homeomorphism h: B^{n}(t) → gU. Moreover,

(t − |h^{−1}(a)|) ∧ (t − |h^{−1}(b)|) ≥ s/LL_{1} ≥ t/c_{1}

with c_{1} = c_{1}(c, c′) = 16c′L^{3}L_{1} . Hence there is L_{2} = L_{2}(c, c′) and an L_{2} - bilipschitz homeomorphism u: B^{n}(t) → B^{n}(t) with u(h^{−1}(a)) = 0, u(h^{−1}(b)) = te_{1}/2. The desired map is then F = h ∘ u^{−1}: B^{n}(t) → gU ⊂ D.

3.10. Lemma. Suppose that f_{k}: D → R^{n} is a sequence of discrete open maps converging locally uniformly to a discrete open map f: D → R^{n}. Then a point a ∈ D is in B_{f} if and only if there are points x_{k} ∈ B_{f_{k}} such that x_{k} → a.

Proof. The ‘if’ part is given in [MR, 3.2]. To prove the converse, it suffices to show that if each f_{k} is a local homeomorphism, then |i(a, f)| = 1.

Choose r > 0 such that U(a, f, 3r) is a normal neighborhood of a; see 3.1. Next choose k ∈ N such that |f_{k}(x) − f(x)| < r/2 for all x ∈ \overline{U}(a, f, r). Then f_{k}(a) ∈ B(f(a), r/2). Let V_{k} be the a-component of f_{k}^{−1}B(f(a), 2r). Then V_{k} does not meet ∂U(a, f, 3r), and hence V_{k} ⊂ U(a, f, 3r). Moreover, V_{k} is a normal domain of f_{k}, and hence f_{k} defines a covering map of V_{k} onto B(f(a), 2r). Since B(f(a), 2r) is simply connected, this map is a homeomorphism. Since U(a, f, r) ⊂ V_{k}, we have

i(a, f) = μ(f(a), f, U(a, f, r)) = μ(f_{k}(a), f_{k}, U(a, f, r)) = ±1;

see [RR, Th. 6, p. 131].

3.11. Theorem. Suppose that D ⊂ R^{n} is a c-uniform domain and that f: D → R^{n} is a K -quasiregular map with N(f) ≤ N < ∞. Then the domain D \ B_{f} is c′ -uniform with c′ = c′(c, N, K, n).

Proof. We show that the domain G = D \ B_{f} satisfies the conditions of 3.8. From the definitions it easily follows that a c-uniform domain is 2c-plump. Hence G is c′ -plump with c′ = c′(c, N, K, n) by 3.4.

It suffices to show that there is c_{0} = c_{0}(c, N, K, n) such that each 8c′ -pair (a, b) in G can be joined by an arc γ such that

δ_{G}(a) ∧ δ_{G}(b) ≤ c_{0} dist (γ, ∂G), diam γ ≤ c_{0}|a − b|.

Assume that this is false for some (c, N, K, n). Then there is a sequence of K - quasiregular maps f_{k}: D_{k} → R^{n} such that N(f_{k}) ≤ N, the domains D_{k} ⊂ R^{n} are c-uniform, and there are 8c′ -pairs (a_{k}, b_{k}) in G_{k} = D_{k} \ B_{f_{k}} such that for any arc γ joining a_{k} and b_{k} in G_{k} we have

(3.12)  δ_{k}(a_{k}) ∧ δ_{k}(b_{k}) > k dist (γ, ∂G_{k})  or  diam γ > k|a_{k} − b_{k}|;

here δ_{k}(x) = dist (x, ∂G_{k}).

Setting r_{k} = δ_{k}(a_{k}) ∧ δ_{k}(b_{k}) and t_{k} = |a_{k} − b_{k}| we have

r_{k} ≤ t_{k} ≤ 8c′r_{k} ≤ 8c′(dist (a_{k}, ∂D_{k}) ∧ dist (b_{k}, ∂D_{k})).

By 3.9 there is L = L(c, c′) and an L-bilipschitz map F_{k}: B^{n}(t_{k}) → D_{k} such that F_{k}(0) = a_{k}, F_{k}(t_{k}e_{1}/2) = b_{k}. Define g_{k}: B^{n} → R^{n} by g_{k}(x) = f_{k}(F_{k}(t_{k}x)). Then g_{k} is K_{1} -quasiregular with K_{1} = K_{1}(K, c, c′, n), and N(g_{k}) ≤ N. By auxiliary similarities we can normalize the situation so that

g_{k}(0) = 0, max{|g_{k}(x)| : |x| ≤ 1/2} = 1.

Applying 2.5 and passing to a subsequence we may assume that the sequence (g_{k}) converges locally uniformly in B^{n} to a nonconstant K_{1} -quasiregular map g: B^{n} → R^{n}.

Since f_{k} is locally injective in B(a_{k}, r_{k}), and since t_{k} ≤ 8c′r_{k}, the map g_{k} is locally injective in B^{n}(1/8c′L) for each k. Hence also g | B^{n}(1/8c′L) is locally injective, which implies that 0 ∉ B_{g} by 3.10. Similarly we obtain e_{1}/2 ∉ B_{g}. Since B^{n} \ B_{g} is connected, we can join 0 and e_{1}/2 by an arc β in B^{n} \ B_{g}. Setting λ = dist (β, S^{n−1} ∪ B_{g}) we have dist (β, B_{g_{k}}) ≥ λ/2 for large k by [MR, 3.2]. The arc γ_{k} = F_{k}[t_{k}β] joins a_{k} and b_{k} in D_{k}. Since

dist (γ_{k}, ∂G_{k}) ≥ \frac{t_{k}λ}{2L} ≥ \frac{r_{k}λ}{2L},

for large k, the first inequality of (3.12) fails for large k. Since diam γ_{k} ≤ Lt_{k}diam β ≤ 2Lt_{k}, the second inequality of (3.12) is not true for large k, and we have reached a contradiction.

3.13. Remark. Theorem 3.11 was proved in [MV, 4.25–4.26] for maps of bounded length distortion. These maps form a proper subclass of the maps considered in 3.11. The case of quasiregular maps is more complicated, since a sequence of maps of L-bounded distortion never converges to a constant.

3.14. The set fB_{f}. Suppose that f: D → R^{n} is a K -quasiregular map with N(f) ≤ N < ∞. Without further restrictions, very little can be said about the set fD \ fB_{f}. It need not be open, and if it is open, in need not be plump, even if D and fD are plump.

However, if D = R^{n}, then f is a closed map onto R^{n}, and we prove in 3.16 that R^{n} \ fB_{f} is a uniform domain. For maps of bounded distortion, this was proved in [MV, 4.25]. A local version is given in 3.17. Both results are corollaries of the more general Theorem 3.15.

3.15. Theorem. Suppose that f: D → R^{n} is a K -quasiregular map and that U ⊂ D is a domain such that

(1) B(U, 2diam U) ⊂ D,

(2) fU is a ball or fU = R^{n},
(3) f defines a proper map U → fU with N(f | U) ≤ N < ∞.

Then fU \ f[B_{f} ∩ U] is a c-uniform domain with c = c(N, K, n).

Recall that B(A, r) denotes the r-neighborhood {x ∈ R^{n} : dist (x, A) < r} of A. Before giving the proof we remark that the conditions of 3.15 hold if U = D = R^{n} and N(f) ≤ N. If D ≠ R^{n}, then (1) implies that U is bounded with \overline{U} ⊂ D. Then (3) means that U is a normal domain of f. In view of the discussion in 3.1, we obtain the following two corollaries of 3.15.

3.16. Theorem. If f: R^{n} → R^{n} is a K -quasiregular map with N(f) ≤ N < ∞, then R^{n} \ fB_{f} is a c-uniform domain with c = c(N, K, n).

3.17. Theorem. Suppose that f: D → R^{n} is K -quasiregular and nonconstant. Then for each x ∈ D with i(x, f) = N ≥ 2 there is r_{0} > 0 such that for 0 < r ≤ r_{0} , U = U(x, f, r) is a normal neighborhood of x, and fU \ fB_{f|U} = B(f(x), r) \ f[U ∩ B_{f}] is a c-uniform domain with c = c(N, K, n).

3.18. Proof of Theorem 3.15. Part 1. We show that the domain G = fU \ f[B_{f} ∩ U] is c-plump with c = c(N, K, n). Assume that this is false for some (N, K, n). Then, for each k ∈ N we can find a K -quasiregular map f_{k}: D_{k} → R^{n} and a domain U_{k} ⊂ D_{k} such that:

(i) Conditions (1)–(3) hold with D = D_{k}, f = f_{k}, U = U_{k}.

(ii) The domain G_{k} = fU_{k} \ f_{k}[B_{f_{k}} ∩ U_{k}] is not k -plump.

By (ii), for each k ∈ N there are y′_{k} ∈ \overline{G}_{k} and s_{k} > 0 such that G_{k} \ B(y′_{k}, s_{k}) ≠ ∅ and such that

(3.19)  B(z, s_{k}/k) ⊄ G_{k}

for every z ∈ \overline{B}(y′_{k}, s_{k}). By (2), there is a ball V′_{k} = B(y_{k}, s_{k}/3) with \overline{V} ′_{k} ⊂ B(y′_{k}, s_{k}) ∩ fU_{k}. The set U_{k} ∩ f_{k}^{−1}(y_{k}) is nonempty and contains at most N points, which we number as a_{1k}, …, a_{Nk}, using repetition if necessary. Let V_{jk} be the a_{jk} -component of f_{k}^{−1}V′_{k}. Then V_{jk} is a normal domain of f_{k}, and

(3.20)  f_{k}V_{jk} = V′_{k},  U_{k} ∩ f_{k}^{−1}V′_{k} = V_{1k} ∪ ⋯ ∪ V_{Nk};

see 3.1.

Set t_{jk} = max\{|x − a_{jk}| : x ∈ \overline{V}_{jk}\}, and define similarities S_{jk} and T_{k} of R^{n} by

S_{jk}(x) = (x − a_{jk})/t_{jk}, T_{k}(x) = 3(x − y_{k})/s_{k}.

From (3) it follows that B(a_{jk}, 2t_{jk}) ⊂ D_{k}. Hence we can define K -quasiregular maps g_{jk}: B^{n}(2) → R^{n} by g_{jk} = T_{k} ∘ f_{k} ∘ S_{jk}^{−1} | B^{n}(2). Setting W_{jk} = S_{jk}V_{jk} we have g_{jk}W_{jk} = B^{n}.

Applying 2.4 with r = 1, R = 2, r′ = 1, A(g_{jk}) = \overline{W}_{jk}, and passing successively N times to subsequences, we may assume that for each j = 1, …, N, the sequence (g_{jk})_{k∈N} converges locally uniformly in B^{n}(2) to a nonconstant K -quasiregular map g_{j}: B^{n}(2) → R^{n} with g_{j}(0) = 0. Set

(3.21)  F_{j} = g_{j}[B_{g_{j}} ∩\overline{B}^{n}],  F = F_{1} ∪ ⋯ ∪ F_{N}.

Since F is a compact set with empty interior, we can find a ball B(w, λ) ⊂ B^{n} \F. From 3.10 it follows that there is k_{0} ∈ N such that B(w, λ/2) does not meet g_{jk}[B_{g_{jk}} ∩\overline{B}^{n}] whenever k ≥ k_{0} and 1 ≤ j ≤ N. For z = T_{k}^{−1}(w) we then have B(z, λs_{k}/6) ⊂ G_{k} for k ≥ k_{0} . By (3.19), this gives a contradiction for large k.

Part 2. Let c = c(N, K, n) be the number given by Part 1. By 3.8, it suffices to find a number c_{0} = c_{0}(N, K, n) such that each 8c-pair (y, z) in G can be joined by an arc γ with the properties

δ_{G}(y) ∧ δ_{G}(z) ≤ c_{0} dist (γ, ∂G), diam γ ≤ c_{0}|y − z|.

We shall modify the proof of Part 1. Assume that c_{0} does not exist for some (N, K, n). Then, for each k ∈ N we can find a K -quasiregular map f_{k}: D_{k} → R^{n}, a domain U_{k} ⊂ D_{k}, and an 8c-pair (y_{k}, z_{k}) in G_{k} = fU_{k} \ f_{k}[B_{f_{k}} ∩ U_{k}] such that:

(i) Conditions (1)–(3) hold with D = D_{k}, f = f_{k}, U = U_{k}.

(ii) If γ is an arc joining y_{k} and z_{k} in G_{k}, then

δ_{G_{k}}(y_{k}) ∧ δ_{G_{k}}(z_{k}) ≥ k dist (γ, ∂G_{k}) or diam γ ≥ k|y_{k} − z_{k}|.

Set

q = 1/9c, V′_{k} = B([y_{k}, z_{k}], q|y_{k} − z_{k}|).

Since (y_{k}, z_{k}) is an 8c-pair in G_{k}, we have \overline{V} ′_{k} ⊂ fU_{k}, and the set U_{k} ∩ f_{k}^{−1}(y_{k}) contains precisely N points a_{1k}, …, a_{Nk}. For each j = 1, …, N, we let V_{jk} denote the a_{jk} -component of f_{k}^{−1}V′_{k}. Then V_{jk} is a normal domain of f_{k}, and (3.20) holds. It is possible that V_{ik} = V_{jk} for some i ≠ j.

Set t_{jk} = max\{|x − a_{jk}| : x ∈ \overline{V}_{jk}\}, and choose similarities S_{jk} and T_{k} of R^{n} such that

S_{jk}(x) = (x − a_{jk})/t_{jk},  T_{k}(y_{k}) = 0,  T_{k}(z_{k}) = e_{1}.

By (1) we again have B(a_{jk}, 2t_{jk}) ⊂ D_{k}, and we can define K -quasiregular maps g_{jk}: B^{n}(2) → R^{n} by g_{jk} = T_{k} ∘ f_{k} ∘ S_{jk}^{−1} | B^{n}(2). Setting W_{jk} = S_{jk}V_{jk} and W′ = B([0, e_{1}], q) we have

max\{|x| : x ∈ \overline{W}_{jk}\} = 1,  g_{jk}W_{jk} = W′,  g_{jk}(0) = 0.

Applying 2.4 with r = 1, R = 2, r′ = 1 + q, A(g_{jk}) = \overline{W}_{jk}, and passing successively to subsequences, we may assume that for each j = 1, …, N, the sequence (g_{jk})_{k∈N} converges locally uniformly in B^{n}(2) to a nonconstant K - quasiregular map g_{j}: B^{n}(2) → R^{n} with g_{j}(0) = 0.

Define F_{j} and F as in (3.21). We show that dist ({0, e_{1}}, F) ≥ q. Assume, for example, that there is u ∈ B_{g_{j}} ∩ B^{n} with |g_{j}(u) − e_{1}| < q. By 3.10 we can find a sequence of points u_{k} ∈ B_{g_{jk}} converging to u. For large k we have |g_{jk}(u) − e_{1}| < q. Since T^{−1}(g_{jk}(u_{k})) = f_{k}(S_{jk}^{−1}(u_{k})) ∈ f_{k}[B_{f_{k}} ∩ U_{k}], this gives the contradiction δ_{G_{k}}(z_{k}) < |y_{k} − z_{k}|/8c.

Since F is a compact set with dim F ≤ n − 2, we can join 0 and e_{1} by an arc α ⊂ W′ \ F. Set λ = dist (α, F ∪ ∂W′) > 0. By 3.10 we can find k_{0} ≥ 2 such that dist (α, g_{jk}[B_{g_{jk}} ∩\overline{B}^{n}]) ≥ λ/2 whenever k ≥ k_{0} and 1 ≤ j ≤ N. Let k ≥ k_{0} . The arc γ = T_{k}^{−1}α joins y_{k} and z_{k} in V′_{k}, and

dist (γ, f_{k}[B_{f_{k}} ∩ V_{jk}] ∪ ∂V′_{k}) ≥ λ|y_{k} − z_{k}|/2

for all 1 ≤ j ≤ N. Since

V′_{k} ∩ f_{k}[B_{f_{k}} ∩ U_{k}] = ⋃_{j=1}^{N} f_{k}[B_{f_{k}} ∩ V_{jk}],

we obtain dist (γ, ∂G_{k}) ≥ λ|y_{k} − z_{k}|/2. Since

diam γ ≤ diam V′_{k} = (1 + 2q)|y_{k} − z_{k}| < 2|y_{k} − z_{k}|,

it follows from (ii) that

k dist (γ, ∂G_{k}) ≤ δ_{G_{k}}(y_{k}) ∧ δ_{G_{k}}(z_{k}) ≤ |y_{k} − z_{k}|,

where the second inequality follows from the property r_{G_{k}}(y_{k}, z_{k}) ≥ 1 of a c-pair; see 3.7. Hence

kλ|y_{k} − z_{k}|/2 ≤ |y_{k} − z_{k}|,

which gives a contradiction for large k.

4. Invariance of NUD and porous sets

4.1. Terminology. A closed set F ⊂ R^{n} is a c-nullset for uniform domains or briefly c-NUD if int F = ∅ and if R^{n} \ F is a c-uniform domain.

Let D ⊂ R^{n} be a domain, let a, b ∈ \overline{D} and let c ≥ 1. We say that a continuum α ⊂ \overline{D} containing a and b satisfies the c-uniformity conditions in D if

diam α ≤ c|a − b|, |x − a| ∧ |x − b| ≤ c dist (x, ∂D)

for all x ∈ α. This implies that α ∩ ∂D ⊂ {a, b}.

If D is c-uniform, it follows from the definition in 3.1 that each pair of distinct points a, b ∈ D can be joined by an arc satisfying the c-uniformity conditions in D. A simple limiting process involving Ascoli’s theorem shows that this is true for all a, b ∈\overline{D}.

Conversely, if each pair of points in a domain D can be joined by a continuum with the c-uniformity properties in D, then D is c′ -uniform with c′ = c′(c, n); see [V4, 2.11].

Suppose that f: R^{n} → R^{n} is K -quasiregular with N(f) ≤ N < ∞. Then B_{f} and fB_{f} are c_{0} -NUD with c_{0} = c_{0}(N, K, n) by 3.11 and 3.16. In this section we show that if F ⊂ R^{n} is c-NUD, then fF and f^{−1}F are c′ -NUD with c′ = c′(c, N, K, n). Similar results hold for porosity.

4.2. Lemma. Suppose that f: B^{n} → R^{n} is a K -quasiregular local homeomorphism with N(f) ≤ N < ∞. Then f is injective in a ball B^{n}(ψ) with ψ = ψ(N, K, n) > 0.

Proof. For n ≥ 3, [MRV3, 2.3] gives the stronger result where ψ = ψ(K, n). For n = 2, one can make use of factorization and basic properties of quasiconformal maps to reduce the question to the case where f is complex analytic. This case follows from the results of C. Pommerenke [Po1, Satz 1.3 and Lemma 1.3].

We give an alternative proof, based on Theorem 2.4, which is valid for all dimensions n ≥ 2. We may assume that f(0) = 0. With the notation of 3.1, we let r_{0}(f) be the supremum of all r > 0 such that U(0, f, r) ⊂ B^{n}(1/2). Clearly 0 < r_{0}(f) < ∞. Replacing f by f/r_{0}(f) we may assume that r_{0}(f) = 1. For 0 < r < 1, f maps U(0, f, r) homeomorphically onto B^{n}(r) by [MRV3, 2.2]. It follows that f maps V (f) = ⋃{U(0, f, r) : r < 1} onto B^{n} and that V (f) = U(0, f, 1). Hence it suffices to find ψ = ψ(N, K, n) > 0 such that B^{n}(ψ) ⊂ V (f).

Let ℱ = ℱ (N, K, n) be the family of all K -quasiregular local homeomorphisms g: B^{n} → R^{n} such that N(g) ≤ N, g(0) = 0, and r_{0}(g) = 1. Then ℱ satisfies the conditions of Theorem 2.4 with r = 1/2, R = 1, r′ = 1, A(g) = \overline{V (g)}. Indeed, g is injective in A(g) by [MRV3, 2.2], and hence in a neighborhood of A(g) by [Zo, p. 422]. By the definition of r_{0}(g), the continuum A(g) meets S^{n−1}(1/2).

By Theorem 2.4, ℱ is a normal family and hence equicontinuous. Consequently, there is ψ > 0 such that gB^{n}(ψ) ⊂ B^{n}(1/2) for all g ∈ ℱ. This implies that B^{n}(ψ) ⊂ V (g), and the lemma is proved.

4.4. Theorem. Suppose that f: R^{n} → R^{n} is K -quasiregular with N(f) ≤ N < ∞, and that F ⊂ R^{n} is c-NUD. Then f^{−1}F is c′ -NUD with c′ = c′(c, N, K, n).

Proof. Let a, b ∈ R^{n}, a ≠ b. Write λ(x) = |x − a| ∧ |x − b|. By 4.1, it suffices to find a continuum β containing a and b such that

(4.5)  diam β ≤ c′|a − b|,  λ(x) ≤ c′ dist (x, f^{−1}F)

for all x ∈ β.

The set B_{f} is c_{0} -NUD with c_{0} = c_{0}(N, K, n) by 3.11. By 4.1, we can join a and b by an arc α such that

(4.6)  diam α ≤ c_{0}|a − b|,  λ(x) ≤ c_{0} dist (x, B_{f})

for all x ∈ α.

Let ψ = ψ(N, K, n) be the number given by 4.2, and set q = q(N, K, n) = ψ/6c_{0} . Then q ≤ 1/6. Orient α from a to b. Pick x_{0} ∈ α with |x_{0}−a| = |x_{0}−b|. Define a sequence of successive points x_{0}, x_{1}, … of α such that x_{j+1} is the last point of α with |x_{j+1} − x_{j}| = qλ(x_{j}). Similarly, define x_{−1}, x_{−2}, … such that x_{−j−1} is the first point of α with |x_{−j−1} − x_{−j}| = qλ(x_{−j}), j ≥ 0. The sequence x_{1}, x_{2}, … converges to a point b′ ∈ α. Since qλ(x_{j}) = |x_{j} − x_{j+1}| → 0, we have b′ = b. Similarly x_{j} → a as j → −∞. Since q ≤ 1/6, we easily see that

(4.7)  \frac{5}{6}λ(x_{j−1}) ≤ λ(x_{j}) ≤ \frac{6}{5}λ(x_{j−1}),  \{x_{j−1}, x_{j+1}\} ⊂\overline{B}(x_{j}, \frac{6}{5}qλ(x_{j}))

for all j ∈ Z.

From (4.6) it follows that dist (x_{j}, B_{f}) ≥ λ(x_{j})/c_{0} for all j ∈ Z. By 4.2, f | B(x_{j}, r) is injective, where r = ψλ(x_{j})/c_{0} = 6qλ(x_{j}). By [V3, 2.4], f is η -quasisymmetric in B(x_{j}, 3qλ(x_{j})) with η = η_{K,n}. We let c_{1}, c_{2}, … denote constants c_{j} ≥ 1 depending only on (c, K, n).

Fact 1. If 0 < t ≤ 3qλ(x_{j}), then B(x_{j}, t) \ f^{−1}F is a c_{1} -uniform domain. Since uniformity is quantitatively preserved by quasisymmetric maps, the

domain fB(x_{j}, t) is c_{2} -uniform. Since F is c-NUD, the domain fB(x_{j}, t) \ F is c_{3} -uniform by [V4, 5.4], and Fact 1 follows by quasisymmetry.

By (4.7) and Fact 1, we can join x_{j−1} and x_{j} by an arc α_{j} satisfying the c_{1} -uniformity conditions in B(x_{j−1}, \frac{6}{5}qλ(x_{j−1}))\f^{−1}F. Then α_{j} ⊂ B(x_{j}, r) with

r = |x_{j−1} − x_{j}| + \frac{6}{5}qλ(x_{j−1}) ≤ \frac{6}{5}qλ(x_{j}) + (\frac{6}{5})^{2}qλ(x_{j}) < 3qλ(x_{j}).

Pick y_{j} ∈ α_{j} with |y_{j} − x_{j−1}| = |y_{j} − x_{j}|. Then {y_{j}, y_{j+1}} ⊂ B(x_{j}, 3qλ(x_{j})). By Fact 1, we can thus join y_{j} and y_{j+1} by an arc β_{j} satisfying the c_{1} -uniformity conditions in the domain G_{j} = B(x_{j}, 3qλ(x_{j})) \ f^{−1}F. Let β be the union of {a, b} and all continua β_{j}, j ∈ Z. We show that β is the desired continuum.

Since diam β_{j} ≤ 6qλ(x_{j}) → 0 as j → ±∞, β is indeed a continuum. If x ∈ β_{j}, then

dist (x, α) ≤ |x − x_{j}| < 3qλ(x_{j}) ≤ \frac{1}{2}diam α.

This and (4.6) yield the first inequality of (4.5) with c′ = 2c_{0} .

To prove the second inequality, assume that x ∈ β_{j}. Then

(4.8)  λ(x) ≤ λ(x_{j}) + |x − x_{j}| ≤ λ(x_{j}) + 3qλ(x_{j}) ≤ \frac{3}{2}λ(x_{j}).

If j ≥ 1, then G_{j} contains the ball B(y_{j}, t) with

t = |y_{j} − x_{j}|/c_{1} ≥ |x_{j−1} − x_{j}|/2c_{1} = qλ(x_{j−1})/2c_{1} > qλ(x_{j})/3c_{1}

by (4.7). Moreover, G_{j} contains B(y_{j+1}, t′) with

t′ = |y_{j+1} − x_{j}|/c_{1} ≥ |x_{j} − x_{j+1}|/2c_{1} = qλ(x_{j})/2c_{1} > qλ(x_{j})/3c_{1}.

Similar arguments show that these estimates hold also for j ≤ 0. By the choice of β_{j} we thus obtain dist (x, f^{−1}F) ≥ qλ(x_{j})/6c_{1}^{2} . By (4.8), this gives the second inequality of (4.5) with c′ = 9c_{1}^{2}/q.

4.9. Lemma. Suppose that f: R^{n} → R^{n} is K -quasiregular with N(f) = N < ∞, and that V ⊂ R^{n} \ fB_{f} is a simply connected c-uniform domain. Then f^{−1}V has precisely N components V_{1}, …, V_{N}, and f defines η -quasisymmetric homeomorphisms f_{j}: V_{j} → V , 1 ≤ j ≤ N, with η depending only on c, N, K, n. Moreover, the domains V_{j} are c′ -uniform with c′ = c′(c, N, K, n).

Proof. The map f is closed and proper by 3.1. Arguing as in [MRV3, 2.2] we see that for each component V_{j} of f^{−1}V , f defines a covering map f_{j}: V_{j} → V . Since V is simply connected, f_{j} is a homeomorphism. To prove the quasisymmetry of f_{j}, we consider a triple (x, y, z) of distinct points in V_{j} with |x − y| ≤ |x − z|. By [V5, 2.9], it suffices to show that

(4.10)  |f(x) − f(y)| ≤ H|f(x) − f(z)|

with H = H(c, N, K, n).

Since V is c-uniform, there is an arc α ⊂ V joining f(x) and f(z) such that diam α ≤ c|f(x) − f(z)|. We may assume that |f(x) − f(y)| > c|f(x) − f(z)|. Define β: [1, ∞) → R^{n} by β(t) = f(x) + t(f(y) − f(x)). Let β^{∗} be a maximal lift of β, starting at y. Then β^{∗} is unbounded. Let Γ be the family of all paths joining α^{∗} = f_{j}^{−1}α and |β^{∗}|. Since |x − y| ≤ |x − z|, a standard estimate gives the lower bound M(Γ) ≥ b_{n} > 0. Since α ⊂\overline{B}(f(x), c|f(x) − f(z)|), we have

M(fΓ) ≤ ω_{n−1}(log \frac{|f(x) − f(y)|}{c|f(x) − f(z)|})^{1−n}.

Since M(Γ) ≤ KNM(fΓ), these estimates yield (4.10). Hence f | V_{j} is η -qua-sisymmetric, and the rest of the lemma follows from the quasisymmetric invariance of uniform domains.

4.11. Lemma. Suppose that D ⊂ G ⊂ R^{n} are domains, that F is closed in G with int F = ∅, and that D and G \ F are c-uniform domains. Then D \ F is a c_{1} -uniform domain with c_{1} = c_{1}(c, n).

Proof. The case G = R^{n} was proved in [V4, 5.4], but the same proof is valid in the general case.

4.12. Corollary. Suppose that D ⊂ R^{n} is a domain, that F_{1}, …, F_{N} are closed in D with int F_{j} = ∅, and that each D \ F_{j} is a c-uniform domain. Then D \ ⋃_{j} F_{j} is a c_{1} -uniform domain with c_{1} = c_{1}(c, N, n).

4.13. Theorem. Suppose that f: R^{n} → R^{n} is K -quasiregular with N(f) = N < ∞, and that F ⊂ R^{n} is c-NUD. Then fF is c′ -NUD with c′ = c′(c, N, K, n).

Proof. The basic idea is the same as in 4.4 and in [V4, 5.4]. Let a, b ∈ R^{n}, a ≠ b. Since fB_{f} is c_{0} -NUD with c_{0} = c_{0}(N, K, n) by 3.16, there is an arc α joining a and b such that

diam α ≤ c_{0}|a − b|, λ(x) ≤ c_{0} dist (x, fB_{f})

for all x ∈ α, where λ(x) = |x − a| ∧ |x − b| as in 4.4. Let c_{1}, c_{2}, … denote constants c_{j} ≥ 1 depending only on (c, N, K, n), and set q = 1/6c_{0} . Choose x_{0} ∈ α with |x_{0} − a| = |x_{0} − b|, and define the points x_{j} ∈ α, j ∈ Z, as in the proof of 4.4. Then (4.7) is again valid for all j ∈ Z.

Fact 1. If 0 < t ≤ 6qλ(x_{j}), then B(x_{j}, t) \ fF is a c_{1} -uniform domain. By 4.9, the set f^{−1}B(x_{j}, t) has N components V_{j1}, …, V_{jN}, and each V_{jk} is

a c_{2} -uniform domain. Moreover, f | V_{jk} is an η -quasisymmetric homeomorphism onto B(x_{j}, t) with η = η_{N,K,n}. Since F is c-NUD, each V_{jk} \ F is c_{3} -uniform. Hence f[V_{jk} \ F] = B(x_{j}, t) \ f[F ∩ V_{jk}] is c_{4} -uniform by quasisymmetry. Fact 1 follows now from 4.12.

The proof can now be completed as in 4.4. The points x_{j−1} and x_{j} are joined by an arc α_{j} satisfying the c_{1} -uniformity conditions in B(x_{j−1}, \frac{6}{5}λ(x_{j−1})) \ fF. Then choose y_{j} ∈ α_{j} with |y_{j} − x_{j−1}| = |y_{j} − x_{j}| and join y_{j} to y_{j+1} by an arc β_{j} satisfying the c_{1} -uniformity conditions in B(x_{j}, 3λ(x_{j})) \ fF. The desired continuum β from a to b is then obtained as the union of all β_{j} and of {a, b}.

4.14. Corollary. If f: R^{n} → R^{n} is K -quasiregular with N(f) ≤ N < ∞, then f^{−1}fB_{f} is c-NUD with c = c(N, K, n).

Proof. This follows from 3.16 and 4.13.

4.15. Theorem. Suppose that f: R^{n} → R^{n} is a K -quasiregular map with N(f) ≤ N < ∞, and that F ⊂ R^{n} is c-porous. Then fF and f^{−1}F are c_{1} -porous with c_{1} = c_{1}(c, N, K, n).

Proof. The idea of the proof is somewhat similar to that in 4.4 and in 4.13, but the present case is much easier. The proofs make use of the porosity of B_{f} and fB_{f}, the quasisymmetric invariance of plumpness, and the fact that the union of two porous sets is porous. The details are omitted.

5. The branch set and the local index

Let f: D → R^{n} be a K -quasiregular map, n ≥ 3, and let a ∈ B_{f}. In [MRV3, 4.4] it was proved that if D\B_{f} contains an open cone with vertex at a and opening angle α, then the local index i(a, f) has an upper bound i(a, f) ≤ N(α, K, n). See also [S1, 3.4 and 4.3].

The direct converse of this result is false. In 5.4 below we give an example of a quasiregular map f: R^{3} → R^{3} with N(f) = 2 such that B_{f} meets every half open line segment (0, y], y ≠ 0.

However, we show that replacing the cone by a curvilinear cone we obtain a condition that is both necessary and sufficient for the existence of such a bound for the local index. This means that we can estimate i(a, f) from above and from below purely in terms of K, n and B_{f}; see 5.2.

5.1. Theorem. Suppose that f: D → R^{n} is K -quasiregular and nonconstant with n ≥ 3. Let a ∈ D, N ≥ 2, and c ≥ 1. Then the following conditions are (K, n)-quantitatively equivalent:

(1) i(a, f) ≤ N,

(2) there is an arc γ ⊂ D with a as an endpoint such that |x − a| ≤ c dist (x, B_{f}) for all x ∈ γ.

More precisely, (1) implies (2) with c = c(N, K, n), and (2) implies (1) with N = N(c, K, n).

Proof. (1) ⇒ (2): Choose a ball B = B(a, r) ⊂ D with N(f | B) ≤ N. The domain G = B \ B_{f} is c_{0} -uniform with c_{0} = c_{0}(N, K, n) by 3.11. Fix a point b ∈ G. By 4.1, we can join a and b by an arc α satisfying the c-uniformity conditions in B \ B_{f}. Let γ be a subarc of α with endpoint a and contained in B(a, |a − b|/2). Then γ satisfies (2).

(2) ⇒ (1): By [MRV3, 5.2], there is a number M = M(K, n) > 1 such that for all x ∈ D we have lim sup_{r→0} H^{∗}(x, f, r) < M, where H^{∗}(x, f, r) = L^{∗}(x, f, r)/l^{∗}(x, f, r) and L^{∗} and l^{∗} are defined as in the proof of 4.2. Assume that (2) does not imply (1) for some (c, N, K, n). Then there is a sequence of K -quasiregular maps f_{k}: B^{n}(2M) → R^{n} with the following properties:

(i) f_{k}^{−1}{0} = {0}, (ii) i(0, f_{k}) ≥ k, (iii) U_{k} = U(0, f_{k}, 1) is a normal neighborhood of 0, (iv) H^{∗}(0, f_{k}, 1) < M,

(v) e_{1} ∈ ∂U_{k}, and there is an arc γ_{k} ⊂ \overline{U}_{k} joining 0 and e_{1} such that dist (x, B_{f_{k}}) ≥ |x|/c for all x ∈ γ_{k},

(vi) f_{k}(e_{1}) = e_{1} .

From (iv) and (vi) it follows that B^{n}(1/M) ⊂ U_{k} ⊂ B^{n}(M). Passing to a subsequence we may assume that the sequence (γ_{k}) converges to a continuum F in the Hausdorff metric of all nonempty compact subsets of \overline{B}^{n}(M). Then \{0, e_{1}\} ⊂ F ⊂\overline{B}^{n}(M). Let ψ = ψ(K, n) be the local injectivity number given by [MRV3, 2.3], and let D be the union of B^{n}(1/M) and the balls B(x, ψ|x|/2c) over x ∈ F \B^{n}(1/M). Since F is connected, D is a domain. Moreover, D ⊂ B^{n}(2M), and we can define the maps g_{k} = f_{k} | D.

We show that the sequence (g_{k}) is equicontinuous. Since B^{n}(1/M) ⊂ U_{k}, we have |f_{k}(x)| < 1 for all |x| < 1/M and for all k. Hence (g_{k}) is equicontinuous in B^{n}(1/M) by [MRV2, 3.17] or by [Re, p. 220]. Suppose that x ∈ F \ B^{n}(1/M). For each y ∈ γ_{k} \ {0}, f_{k} is injective in B(y, ψ|y|/c) by (v) and by the choice of ψ. Since γ_{k} → F, the maps f_{k} | B(x, ψ|x|/2c) are injective for large k. Since they omit 0 and ∞ by (i), it follows from [V2, 19.3] that (g_{k}) is equicontinuous in B(x, ψ|x|/2c), and hence in D.

Passing to a subsequence we may assume that (g_{k}) converges locally uniformly to a K -quasiregular map g: D → R^{n}. Since g(0) = 0 and g(e_{1}) = e_{1} , g is nonconstant. By [MRV3, 4.5], i(0, g) ≥ lim sup_{k→∞} i(0, g_{k}). This contradicts (ii) and completes the proof.

We give a slightly different formulation of Theorem 5.1.

5.2. Theorem. Suppose that f: D → R^{n} is K -quasiregular and nonconstant with n ≥ 3. For a ∈ D, let u = u(a, B_{f}) denote the infimum of all c ≥ 1 satisfying condition (2) of Theorem 5.1. Then

N_{1}(u, K, n) ≤ i(a, f) ≤ N_{2}(u, K, n) < ∞,

where N_{1}(u, K, n) → ∞ as u → ∞.

Proof. Let c(N, K, n) and N(c, K, n) be the functions given by 5.1. The second inequality of 5.2 holds, for example, with N_{2}(u, K, n) = N(u + 1, K, n).

If (K, n) is a pair such that i(a, f) is bounded by a number M(K, n) for all f and a, then u ≤ c(M(K, n), K, n), and the first inequality of 5.2 is an empty condition. Assume that (K, n) is a pair such that i(a, f) may have arbitrarily large values. Such pairs exist for all n ≥ 3 by [MRV3, 4.9]. By 5.1, c(N, K, n) → ∞ as N → ∞. For t ≥ 1, let N_{1}(t, K, n) be the maximum of all integers m such that c(m, K, n) < t, with N_{1}(t, K, n) = 1 if there are no such integers. From 5.1 it follows that the theorem holds with this function N_{1} .

5.3. Open problem. Is it possible to replace the arc in 5.1 by a sequence of points converging to a? More precisely, suppose that (x_{j}) is a sequence of points in D \ B_{f} converging to a such that |x_{j} − a| ≤ c dist (x_{j}, B_{f}) for all j. Is i(a, f) bounded by a constant N(c, K, n)?

5.4. Theorem. There is a quasiregular map f: R^{3} → R^{3} with N(f) = 2 such that B_{f} meets (0, y] for each y ∈ R^{3} \ {0}.

Proof. The theorem follows from Lemmas 5.5 and 5.11 below.

5.5. Lemma. Let Z ⊂ R^{3} be the line span (e_{3}), and let g: R^{3} → R^{3} be a K -quasiconformal map. Then there is a 4K -quasiregular map f: R^{3} → R^{3} with N(f) = 2 such that B_{f} = gZ.

Proof. Let w: R^{3} → R^{3} be the 4-quasiregular winding map, defined by w(r, φ, z) = (r, 2φ, z) in the cylindrical coordinates. Then f = w ∘ g^{−1} is the desired map.

5.6. Remark. There is also a map f: R^{3} → R^{3} such that B_{f} = fB_{f} = gZ and such that f is of L-bounded length distortion with L = L(K). See [MV, 4.27].

If Q ⊂ R^{3} is a closed ball of radius r and if t > 0, we let Q(t) denote the concentric ball with radius tr.

5.7. Lemma. There is t ∈ (1/2, 1) and a finite family ℬ of disjoint closed balls in B^{3} \\overline{B}^{3}(t) such that if R ⊂ R^{3} is a ray from a point in \overline{B}^{3}(t), then R meets Q(t) for some Q ∈ ℬ.

Proof. A construction for the corresponding result in R^{2} is given in Figure 1 with t = 9/10. The construction in R^{3} is rather similar but somewhat more complicated. We omit the details.

Figure 1

5.8. The Cantor set C. By a parallel similarity we mean a map f: R^{3} → R^{3} of the form f(x) = λx + b, λ > 0, b ∈ R^{3} . Let t > 0 and ℬ = {Q_{1}, …, Q_{m}} be the number and the family of balls given by 5.7. For 1 ≤ j ≤ m, let β_{j} be the parallel similarity with β_{j}\overline{B}^{3} = Q_{j}. These maps define in the familiar way a self similar Cantor set C. More precisely, C is the intersection of the descending sequence of compact sets C_{k}, where C_{1} = Q_{1} ∪ ⋯ ∪ Q_{m}, C_{2} is the union of the balls β_{i}Q_{j}, etc.

5.9. Lemma. Let R ⊂ R^{3} be a ray from a point in \overline{B}^{3}(t). Then R meets C.

Proof. Let k ∈ N. It suffices to show that R meets C_{k}. For k = 1 this follows from 5.7. In fact, R meets Q_{j}(t) for some j. Hence β_{j}^{−1}R meets \overline{B}^{3}(t). By 5.7, there is i such that β_{j}^{−1}R meets Q_{i}(t). It follows that R meets β_{j}Q_{i}(t) ⊂ C_{2} . Proceeding inductively in this manner we obtain the lemma.

5.10. Lemma. There is a quasiconformal map h: R^{3} → R^{3} such that h(x) = x for |x| ≤ t and for |x| ≥ 1, and such that h[te_{3}, e_{3}] contains C.

Proof. Choose disjoint closed balls A_{1}, …, A_{m} in B^{3} \\overline{B}^{3}(t) with centers on the line segment [te_{3}, e_{3}]. For each j = 1, …, m, let α_{j} be the parallel similarity with α_{j}\overline{B}^{3} = A_{j}. Choose a homeomorphism h_{1}: R^{3} → R^{3} such that

h_{1}(x) = x for |x| ≤ t and for |x| ≥ 1,

h_{1}(x) = β_{j}α_{j}^{−1}(x) for x ∈ A_{j}, 1 ≤ j ≤ m,

and such that h_{1} is K -quasiconformal with some K. Next define h_{2}: R^{3} → R^{3} by h_{2} = h_{1} outside the balls A_{j}, and by h_{2} = β_{j}h_{1}α_{j}^{−1} in A_{j}. Iterating the construction in the natural way we obtain a sequence (h_{k}) of K -quasiconformal maps converging to a K -quasiconformal map h: R^{3} → R^{3} with the desired properties.

5.11. Lemma. There is a quasiconformal map g: R^{3} → R^{3} such that gZ meets every line segment (0, y], y ≠ 0.

Proof. We set g(x) = x if x = 0 or if |x| ≥ 1. Let 0 < |x| < 1. Then t^{k+1} ≤ |x| < t^{k} for a unique integer k ≥ 0, and we set g(x) = t^{k}h(x/t^{k}), where h is given by 5.10. Then g is clearly a K -quasiconformal homeomorphism. From 5.9 and 5.10 it follows that g[te_{3}, e_{3}] meets [te, e] for every e ∈ S^{2} . By construction, g[0, e_{3}] meets every (0, y].

5.12. Remarks. 1. If g: R^{3} → R^{3} is bilipschitz, then the Hausdorff dimension of gZ is 1, and hence the set of all e ∈ S^{2} with (0, e] ∩ gZ ≠ ∅ is of area zero. However, there is a bilipschitz map g such that for every r > 0, gZ meets B^{3}(r) ∩ V for each open cone V with vertex at 0; see [LV, 4.11].

2. On the other hand, the map f of 5.4 can be chosen to be of bounded length distortion in view of Remark 5.6.

3. Similar examples exist in R^{n} for all n ≥ 3.

We finally give a result in a direction converse to the cone theorem [MRV3, 4.4]. For y ∈ S^{n−1} and 0 ≤ α ≤ π/2, let C(y, α) denote the open cone {x ∈ R^{n} : x · y > |x| cos α}.

5.13. Theorem. Suppose that n ≥ 3, that f: B^{n} → R^{n} is a nonconstant K -quasiregular map with f(0) = 0, and that 0 < α < 1/2. If for some r_{0} > 0,

B_{f} ∩ C(y, α) ∩ B^{n}(r) \ B^{n}((1 − α)r) ≠ ∅

for all y ∈ S^{n−1} and for all r ∈ (0, r_{0}], then i(0, f) ≥ N_{1}(α, K, n), where N_{1}(α, K, n) → ∞ as α → 0.

Proof. This follows easily either from 3.4 or from 5.2.

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Received 23 September 1997 Received in revised form 5 June 1998