ZORIC’S THEOREM,

MAPPINGS OF FINITE DISTORTION:

ZORIC’S THEOREM,

AND EQUICONTINUITY RESULTS

MIHAI CRISTEA

We generalize some known results from the theory of quasiregular mappings, as Zoric’s theorem and some equicontinuity results. These generalizations hold for a special class of mappings with finite distortion satisfying condition (A), which extends the known class of quasiregular mappings, for which some recent modular and capacity inequalities established in [9] are valid.

AMS 2000 Subject Classification: 30C65.

Key words: Zoric’s theorem, Montel’s theorem, mapping of finite distortion.

1. INTRODUCTION

A mapping f : D → Rⁿ, where D ⊂ Rⁿ is a domain, is said to have finite distortion if the following conditions are satisfied:

1) f ∈W_loc^1,1(D,Rⁿ);

2) the Jacobian determinant J_f(x) is locally integrable;

3) there exists a finite a.e. measurable functionK:D→[0,∞] such that

|f(x)|ⁿ≤K(x)·Jf(x) a.e.

Notice that whenK ∈L^∞(D) we obtain the known class of quasiregular mappings, and we refer the reader to [15], [14] for monographs devoted to this subject. Quasiregular mappings are either constant or open, discrete, satisfy condition (N) and a H¨older condition.

If the dilatation mapK ∈ L^p_loc(D), p > n−1, and f ∈ W_loc^1,n(D,Rⁿ), then it is shown in [11] that a map with finite distortion is open, discrete.

Recently, there were considered in [11], [9], [10], [13] mappings f :D→ Rⁿ of finite distortion for which there exists some smooth, strictly increas- ing A : [0,∞) → [0,∞) with A(0) = 0, lim

t→∞A(t) = ∞ and satisfying the conditions:

(A₀) exp(A(K))∈L¹_loc(D);

(A1)_∞

1 A(t)

t dt=∞;

REV. ROUMAINE MATH. PURES APPL.,52(2007),5, 539–554

(A₂) there exists t₀>0 such thatA(t)·tincreases to infinity fort≥t₀. As in [9], we say that a mapping of finite distortion satisfies condition (A) iff satisfies condition (A₀), (A₁), (A₂). In [7] is shown that such nonconstant mappings are either constant or open, discrete. Our proofs are based on the following basic ingredients, valid for mappingsf :D→Rⁿof finite distortion and satisfying condition (A):

(i) M(f(Γ))≤M_Kn−1(Γ) for every path family Γ from D;

(ii) capf(E) ≤ cap_Kn−1(E), where E = (G, C) is a capacitor with G⊂⊂D;

(iii) cap_Kn−1(B(x, R), B(x, r)) → 0 as r → 0 and R > 0 is kept fixed.

They were established in [9], Corollary 4.2, Corollary 5.2 and Theorem 5.3.

Here, K is the dilatation map of f. We also give in Lemma 1 a condition able to ensure that cap_Kn−1(B(x, R), B(x, r)) → 0 as r > 0 is kept fixed and R→ ∞. This last condition is necessary in the proof of Zoric’s theorem (Theorems 2 and 1).

We generalize Zoric’s theorem for mappings with finite distortion which satisfy condition (A) as follows.

Theorem 2 (Zoric’s theorem). Let n ≥3. Let f :Rⁿ → Rⁿ be a local homeomorphism and a mapping of finite distortion satisfying condition (A) for which there exists r > 0 such that

B(0,r)

exp(A(K(x)))· _|x¹_|2ndx < ∞. Then f is a global homeomorphism.

We give a Picard type theorem in Theorem 4, in connection with a similar result from [9].

Theorem 4 (Picard’s theorem). Let F ⊂ Rⁿ be closed, f : Rⁿ\F → Rⁿ a nonconstant map of finite distortion, satifying condition (A),such that M_Kn−1(F) = 0and there existsr >0such that

B(0,r)

exp(A(K(x)))·_|x¹_|2ndx <

∞. Then capf(Rⁿ\F) = 0.

We also give in Theorem 1 a version of Zoric’s theorem with some “sin- gular” sets K and B, extending some results from [1] and [2]. Our variants of Zoric’s theorem may be put in connection with some recent result from [6] and [11]. However, our results ensure that the map f : Rⁿ → Rⁿ is a homeomorphism, hence we give a complete generalization of Zoric’s theorem.

Indeed, in [6] is proved that if f :Rⁿ → Rⁿ is a local homeomprhism and a mapping of finite distortion ofK(x) dilatation satisfying condition (A), and M_Kn−1(∞) = 0, then there existsE⊂Rⁿ with capE = 0 such thatf :Rⁿ→ Rⁿ\E is a homeomorphism. Our condition

B(0,r)

exp(A(K(x)))·_|x¹_|2ndx <∞

implies that M_Kn−1(∞) = 0, hence our Theorem 2 generalizes the so called Global Homeomorphism Theorem from [6]. Also, in [11] is proved that if f :B(0, r)→Rⁿis a local homeomorphism of finite distortion ofK(x) dilata- tion satisfying conditon (A) with exp(A ◦K)∈L¹(B(0, r)), then there exists δ(n, K, r) such thatf is injective onB(0, δ). The result from [11] is not enough for an arbitrary local homeomorphismf :Rⁿ→Rⁿof finite distortion ofK(x) dilatation and satisfying condition (A) to be a global homeomorphism, as in the classical case when K(x) ≤ K on Rⁿ, i.e., we cannot prove Zoric’s the- orem for mappings of finite distortion using [11]. An example from [8] shows that the fundamental condition

B(0,r)

exp(A(K(x)))· _|x¹_|2ndx < ∞ used in Theorems 2 and 4 is necessary in order to ensure thatf :Rⁿ→Rⁿis a global homeomorphism.

We generalize in Theorem 5 a known equicontinuity result for quasireg- ular mappings as follows.

Theorem 5. Let D ⊂Rⁿ be a domain, M ⊂Rⁿ with capM > 0, W a family of mappings f :D→Rⁿ\M of finite distortion satisfying condition (A) and having the same dilatation map K. Then W is equicontinuous if we consider the Euclidean distance on Dand the chordal distance on Rⁿ.

We see that even in the case where all the mappings from the familyW areK-quasiregular, we improve the classical result established for quasiregular mappings in [15], Corollary 2.7, page 66. Indeed, the exceptional setM which is avoided by all the mapsf ∈W can be to be at most countable and chosen such that capM >0, hence of zero capacity.

We immediately obtain a Montel type theorem for mappings of finite distortion satisfying condition (A), extending in this way Theorem 8.14.1, [4], established in the particular caseA(t) =λt.

Theorem 6 (Montel’s theorem). Let D⊂ Rⁿ be a domain. Let W be a bounded family of mappings f :D→ Rⁿ of finite distortion with the same dilatation map K and satisfying condition (A). Then W is a normal family.

2. PRELIMINARIES

We call open a path q : [0,1) → Rⁿ. A point x ∈ Rⁿ will be called a limit point of q if there exists t_p 1 such that q(t_p) → x. If Γ is a path family in Rⁿ, we define F(Γ) = {ρ : Rⁿ → [0,∞] is a Borel map

|

γρds≥1 for every locally rectifiable γ ∈Γ}.Let nowD⊂Rⁿ be open and ω :D → [0,∞] measurable and finite a.e. If ω : Rⁿ → [0,∞] is defined by

ω(x) =ω(x) if x∈D, ω(x) = 0 if x /∈D, then we define the ω-modulus of Γ byM_ω(Γ) = inf

ρ∈F(Γ)

Rⁿ

ρⁿ(x)·ω(x)dx. For ω ≡1 we obtain the usual modulus M(Γ). If Γ₁, Γ₂are path families inRⁿ, we say that Γ₁>Γ₂if every path from Γ₁ has a subpath in Γ₂. As in the classical case, we can prove that if Γ₁>Γ₂, thenMω(Γ₁)≤Mω(Γ₂). Also, we can prove thatMω

_∞

i=1 Γi

≤ ^∞

i=1Mω(Γi), and if ω₁ ≤ ω₂, then M_ω1(Γ) ≤M_ω2(Γ). For D ⊂ Rⁿ open, E, F ⊂ D, we denote by ∆(E, F, D) the family of all paths, open or not, which join E with F inD.

We say that E = (D, C) is a condenser if C is compact, D is open, C ⊂ D ⊂ Rⁿ. Let ω :D → [0,∞] be measurable and finite a.e. We define theω-capacity of E by cap_ωE = inf

Rⁿ|∇u|ⁿ(x)·ω(x)dx, where u∈C₀^∞(D) and u ≥ 1 on C. If ω = 1 we obtain the usual capacity cap E of E. We see that ifu is a test function for capωE, then ρ =|∇u| ∈ F(ΓE). This implies that Mω(ΓE) ≤ capω(E). Here, ΓE = {γ : [a, b] → D is a path | γ(a) ∈ C and γ has a limit point in ∂D} and from Proposition 10.2 in [15, page 54], we have capE = M(Γ_E). If C ⊂Rⁿ is compact, we say that cap C = 0 if cap(A, C) = 0 for every C⊂A⊂Rⁿ open. By [15, Lemma 2.2, page 64], the definition is independent on the open set A such that C ⊂A. If C ⊂ Rⁿ is arbitrary, we say that capC = 0 if capK= 0 for every compact K ⊂C.

Let now D ⊂ Rⁿ be open, ω : D → [0,∞] measurable and finite a.e., A⊂D a set. We say that A is of zeroω-modulus, and we write Mω(A) = 0, if the ω-modulus of all paths having some limit point in A is zero. If ω ≥1, we see thatM(Γ)≤M_ω(Γ). Hence, ifM_ω(A) = 0 then capA= 0.

If A ⊂D is countable and lim

r→0Mω(∆(B(x, r),B(x, R), D)) = 0 for ev- ery x ∈ A and every 0 < R with B(x, R) ⊂ D, we can prove as in the classical case that Mω(A) = 0. Using Theorem 5.3 in [9], we see that this holds, for instance, if ω = Kⁿ⁻¹, where K : D → [0,∞] is measurable and finite a.e. and for every x ∈ A there exists ρx > 0 such that B(x, ρx) ⊂ D

and

B(x,ρ_x)

exp(A(K(y)))dy <∞, whereAsatisfies conditions (A₁) and (A₂).

Lemma 1 also gives a condition which ensures thatM_Kn−1(∞) = 0.

IfD⊂Rⁿis open andb∈∂D, we defineC(f, b) ={w∈Rⁿ|there exists bp∈D,bp→bsuch thatf(bp)→w}. IfB⊂∂D, we letC(f, B) =

b∈BC(f, b).

If E, F are Hausdorff spaces, f : E → F is a map, p : [0,1] → F is a path, x∈E is such thatf(x) =p(0),we say thatq: [0, a)→E is a maximal lifting ofpfromxwhen q(0) =x, 0< a≤1, f◦q =p|[0, a) andq is maximal with this property. Ifqis defined on [0,1], we say thatq is a lifting ofp. IfE, F are

domains in Rⁿ and f is continuous, open, discrete, then there always exists a maximal lifting. We denote by µ_n the Lebesgue measure in Rⁿ and by q the chordal metric inRⁿ given byq(a, b) =|a−b| ·(1 +|a|²)⁻¹² ·(1 +|b|²)⁻¹² ifa, b ∈ Rⁿ, q(a,∞) = (1 +|a|²)⁻¹2 if a∈ Rⁿ, and by |a−b| the Euclidean distance betweenaand b inRⁿ. We denote by Bq(x, r), respectivelyB(x, r), the ball of center x and radius r if we consider on Rⁿ the chordal metric, respectively the Euclidean metric. In the same way, for a setA⊂Rⁿwe denote byq(A), respectivelyd(A), the diameter ofA. We denote by W_loc^1,p(D,Rⁿ) the Sobolev space of all functionsf :D→Rⁿwhich are locally inL^p(D), together with their first order weak partial derivatives.

IfW is a family of mappingsf :D→Rⁿ, we say that W is bounded if for every compact K ⊂ D there exists M(K) >0 such that |f(x)| ≤ M(K) for everyx∈K and everyf ∈W. IfX, Y are metric spaces andW is a family of mappings f : X → Y, we say that W is equicontinuous at x if for every ε >0 there exists δε > 0 such that d(f(x), f(y)) ≤ ε when d(y, x) ≤ δε for everyf ∈W. We say thatW is equicontinuous if it is equicontinuous at every point x ∈X. We say that W is a normal family if every sequence in W has a subsequence which converges uniformly on the compact subsets of X to a mapf :X →Y.

Lemma 1. Let n ≥ 2. Let D ⊂ Rⁿ be an unbounded domain, K : D → [0,∞] measurable and finite a.e., A : [0,∞) → [0,∞) smooth, strictly increasing, with A(0) = 0, lim

t→∞A(t) =∞,satisfying condition (A1) and (A2) for which there exists r > 0 such that

D∩B(0,r)

exp(A(K(x)))· _|x¹_|2ndx <∞. ThenM_Kn−1(∆(B(0, r)∩D,B(0, s)∩D, D)→0as s→ ∞and r >1is kept fixed.

Proof. Let g : Rⁿ → Rⁿ be defined by g(x) = ^x

|x|² if x ∈ Rⁿ and let Γrs = ∆(B(0, r)∩D,B(0, s)∩D, D), Λrs = ∆(g(D)∩B(0,¹_s), g(D)∩ B(0,¹_r), g(D)) for 1 < r < s. We show that M_Kn−1(Γrs) = M_Kn−1◦g(Λrs).

Letρ ∈F(Γrs) and ρ:Rⁿ →[0,∞] be defined by ρ(x) =ρ(g(x))· |g(x)| if x ∈ D, ρ(x) = 0 if x /∈ D. Then ρ ∈ Γ(Λrs) and M_Kn−1◦g(Λrs) ≤

Rⁿ

ρⁿ(x)· Kⁿ⁻¹(g(x))dx=

D

ρⁿ(g(x))|g(x)|ⁿ·Kⁿ⁻¹(g(x))dx=

D

ρⁿ(g(x))·Kⁿ⁻¹(g(x))·

J_g(x)dx =

g(D)

ρⁿ(y)·Kⁿ⁻¹(y)dy ≤

Rⁿ

ρⁿ(y)·Kⁿ⁻¹(y)dy. This implies that M_Kn−1◦g(g(Γ_rs)) =M_Kn−1◦g(Λ_rs)≤M_Kn−1(Γ_rs). By the same argument, we haveM_Kn−1(Γrs) =M_Kn−1◦g◦g(g(Λrs))≤M_Kn−1◦g(Λrs) and

(1) M_Kn−1(Γrs) =M_Kn−1◦g(Λrs).

Next,

g(D)∩B(0,¹_r)

exp(A(K(g(x))))dx=

g(D)∩B(0,¹_r)

exp(A(K(g(x))))·Jg⁻¹(g(x))·Jg(x)dx=

=

D∩B(0,r)

exp(A(K(y)))·J_g−1(y)dy=

D∩B(0,r)

exp(A(K(y)))· 1

|y|²ⁿdy <∞. Let now Γ_rs = ∆(B(0,¹_s),B(0,¹_r), B(0,¹_r)\B(0,¹_s)) for 1 < r < s and Q : Rⁿ → [0,∞] be defined by Q(x) = K(g(x)) if x ∈ g(D), Q(x) = 1 if x /∈g(D). Since

B(0,¹_r)

exp(A(Q(x)))dx <∞, by Theorem 5.3, page 24, in [9], cap_Qn−1(B(0,¹_r), B(0,¹_s)) → 0 as s → ∞ and r > 0 is kept fixed. We have M_Kn−1◦g(Λ_rs) ≤M_Qn−1(Λ_rs) ≤M_Qn−1(Γ_rs) ≤cap_Qn−1(B(0,¹_r), B(0,¹_s)) and this implies that

(2) M_Kn−1◦g(Λ_rs)→0 ass→ ∞ and r >0 is kept fixed.

Using (1) and (2), the proof is complete.

Lemma 2. Let D ⊂ Rⁿ be open. Let f : D → Rⁿ be a map of finite distortion satisfying condition (A) and K : D → [0,∞] the dilatation map of f. Then K ∈ L^p_loc(D) for every p > 0 and f ∈ W_loc^1,p(D,Rⁿ) for every 1≤p < n.

Proof. Using property (A₂) we see that the function exp(A(t)) grows faster then anyt^p to infinity and therefore K ∈L^p_loc(D) for every 0< p <∞. We use now H¨older’s inequality and the integrability of the Jacobian to see that

B|f(x)|^pdx <∞for every ballB withB ⊂Dand every 1≤p < n.

3. PROOFS OF THE MAIN RESULTS We start with

Theorem 1. Let n ≥3. Let B ⊂ Rⁿ be closed, K ⊂ Rⁿ\B closed in Rⁿ\B with intK = ∅, f :Rⁿ\(K ∪B) → Rⁿ a map of finite distortion of K(x)dilatation satisfying condition (A)and a local homeomorphism such that M_Kn−1(B) = 0, intC(f, K) =∅, C(f, K) closed and Rⁿ\C(f, K) connected.

Let E =K∪B∪f⁻¹(C(f, K)).Suppose that exp(A(K))∈L¹_loc(Rⁿ\K) and that there exists δ₀ >0such that

B(0,δ₀)∩(Rⁿ\(K∪B))

exp(A(K(x)))·_|x¹_|2n <∞.

Then

1)We can extend f to a local homeomorphism around each point x∈B with possibly f(x) = ∞ for some point x ∈ B. Also, extending if necessary the map f to a homeomorphism near ∞, we can lift every path: p[0,1] → Rⁿ\C(f, K) from every point x∈Rⁿ with f(x) =p(0).

2) If C(f, K) is compact, then intE = ∅ and f is injective on Rⁿ\E.

If f can be continuously extended to K, then either f|Rⁿ\K : Rⁿ\K → Rⁿ\C(f, K)orf|Rⁿ\K :Rⁿ\K→ Rⁿ\C(f, K)is a homeomorphism. Finally, if f can be continuously extended to an open map on K, then f :Rⁿ → Rⁿ is a homeomorphism.

Proof. By Theorem 1.1 [13], we can take B =∅. Let Γ = {γ : [0,1) → Rⁿ\K path|∞ ∈Im γ}. By Lemma 1,M_Kn−1(Γ) = 0.Using (i), we see that M(f(Γ))≤M_Kn−1(Γ) = 0, henceM(f(Γ)) = 0 and by Theorem 1 in [2], 1) is proved.

Suppose now that C(f, K) is compact and let r > 0 be such that C(f, K) ⊂ B(0, r), let 0 < r < s and let x ∈ Rⁿ\K be such that f(x) ∈ B(0, s). LetQbe the component off⁻¹(B(0, r)) which containsxand letU be the component off⁻¹(B(0, s)) which contains x. Then U ⊂Q and since B(0, r) is simply connected and f|Q : Q → B(0, r) is a local homeomor- phism which lifts the paths,f|Q :Q→ B(0, r) is a homeomorphism, hence f|U : U → B(0, s) also is a homeomorphism. Since f|∂U : ∂U → S(0, s) is a homeomorphism, Jordan’s theorem implies thatRⁿ\∂U has exactly two components, one of them being U. Since U is homeomorphic to B(0, s), U is unbounded. If U₀ is any other component of f⁻¹(B(0, s)), by the same argument, f|U₀ :U₀ → B(0, s) is a homeomorphism, ∂U₀ bounds a Jordan domain, and U₀ is the unbounded component ofRⁿ\∂U₀. Let α >0 be such that ∂U ∪∂U₀ ⊂ B(0, α). Then B(0, α) ⊂ U ∩U₀, hence U = U₀ and we obtain thatf⁻¹(B(0, s)) has a single component U on which f is injective, K⊂U and U =f⁻¹B(0, s).

Since f is a local homeomorphism on Rⁿ\K and int C(f, K) = ∅, we have int E = ∅. Let x₁, x₂ ∈ Rⁿ\E be such that f(x₁) = f(x₂) = y. Let

z ∈ B(0, s) and p : [0,1] → Rⁿ\C(f, K) a path such that p(0) = y and

p(1) =z. We can liftp from x₁ and x₂ and find some paths qi : [0,1]→ Rⁿ such that q_i(0) = x_i, f ◦ q_i = p, i = 1,2. Since q_i(1) ∈ f⁻¹(B(0, s)) = U, f(q_i(1)) =p(1), i= 1,2 and we proved thatf is injective onU, we obtain thatq₁(1) =q₂(1).We now use the property of the uniqueness of path lifting for local homeomorphisms to conclude thatx₁ =x₂. We therefore proved that f is injective onRⁿ\E.

Suppose now that f can be continuously extended to K and let a∈Rⁿ be a point such that f is open at a. Let us show that {a} = f⁻¹(f(a)).

Indeed, if there existsb=a,b∈Rⁿ such that f(a) =f(b), let U₁∈ V(a) and V ∈ V(f(a)) be such that f(U₁) =V. Sincef is continuous at b, we can find U₂∈ V(b) withU₁∩U₂=∅such thatf(U₂)⊂V, and since intE =∅, we can findα ∈U₂\E. Then we can findβ ∈U₁\E withf(α) =f(β). So, we reached a contradiction, since we proved that f is injective onRⁿ\E. It results now immediately that eitherf|Rⁿ\K:Rⁿ\K →Rⁿ\f(K) orf|Rⁿ\K :Rⁿ\K → Rⁿ\C(f, K) is a homeomorphism. If we additionally suppose that f also is an open map onK, we find thatf|Rⁿ:Rⁿ→Rⁿ is a homeomorphism.

Proof of Theorem 2. We takeK =∅ in Theorem 1.

The following version of Zoric’s theorem seems to be new even for quasire- gular mappings.

Theorem 3. Let n ≥ 3. Let K ⊂ Rⁿ be compact, f : Rⁿ → Rⁿ con- tinuous on Rⁿ and of finite distortion of K(x) dilatation satisfying condition (A) on Rⁿ\K for which there exists r >0 such that f is a local homeomor- phism on B(0, r)and

B(0,r)

exp(A(K(x)))·_|x¹_|2ndx <∞. If fis unbounded, then there exists δ >0 such that f is injective on B(0, δ) while if f is open discrete on K,then f :Rⁿ→Rⁿ is a homeomorphism.

Proof. Suppose first that f is unbounded and let ρ > 0 be such that f(K)⊂B(0, ρ). Letp: [0,1]→B(0, ρ) be defined byp(t) = (1−t)f(x) +ty fort∈[0,1] and some x∈Rⁿ\K and y ∈CB(0, ρ). As in Theorem 1 in [2], we try to liftpfrom x. Let 0< a≤1 and letq : [0, a)→Rⁿ\K be a maximal lifting ofp from x withf ◦q=p|[0, a). As in our Theorem 1 and Theorem 1 in [2] we can find spheresS(p(a), r) and sets Qr which bound some Jordan domains Dr such that f|Qr : Qr → S(p(a), r) is a homeomorphism for a.e.

r >0, and we can suppose that B(0, ρ)∩B(p(a), r) =∅ for a.e. 0< r≤r₀. As in Theorem 1 in [2], we have two cases.

In the first case, suppose that D_r ⊂ D_s for 0 < s < r ≤ r₀. Let W_r = _s<r(D_s\D_r) and B_r = B(p(a), r)∪B(0, ρ) for 0 < r ≤r₀. Then B_r is connected for 0 < r ≤ r₀ and let us fix 0 < s < r ≤ r₀. We show that f((Ds\Dr)\K)⊂Br. Indeed, otherwise we can find a point α ∈(Ds\Dr)\K such that f(α) ∈ CBr. Let y ∈ Br and let Γ : [0,1] → Br be a path such that Γ(0) = f(α), Γ(1) = y. If we cannot lift Γ from α, we can find 0< c ≤1 and a maximal lifting γ : [0, c) → Rⁿ\K of Γ from α. Let t_p c be such that γ(t_p) → z. Then z ∈ D_s, hence z = ∞ and z ∈ Q_s∪Q_r∪K, hence f(z) ∈ f(Qs ∪Qr ∪K) ⊂ Br. On the other hand, we have f(z) = f

plim→∞γ(tp)

= lim

p→∞f(γ(tp)) = lim

p→∞Γ(tp) = Γ(c) ∈ Br and we so reach a contradiction.

We see that we can lift Γ from α, hence we can find a path γ : [0,1]→ (D_s\D_r)\K with γ(0) = α and Γ = f ◦γ, hence y = Γ(1) = f(γ(1)) ∈ f((D_s\D_r)\K). Since y was arbitrary chosen in B_r, we proved that B_r ⊂ f((D_s\D_r)\K) ⊂f(D_s), so we reached a contradiction so f(D_s) is compact andBr is unbounded.

We therefore proved that f(Ds\Dr) ⊂ Br for 0 < s < r ≤ r₀. Let A = lim

r→∞W_r. Then W_r is connected and f(W_r) ⊂ B_r for 0 < r ≤ r₀, hence C(f, A) = ∩

r>0f(W_r) is connected and C(f, A) ⊂ B_r for 0 < r ≤ r₀. Since p(a) ∈ C(f, A), we have C(f, A) ⊂ B(p(a), r) for 0 < r ≤ r₀, hence C(f, A) ={p(a)}. This implies thatA∩K=∅and since f is a light map on Rⁿ\K, we have cardA = 1. ThenA={∞} and f maps a neighbourhood of

∞near p(a) and this contradicts the fact that f is unbounded. We therefore proved that the caseDr⊂Ds for 0< s < r≤r₀ cannot hold.

In the second case, we have Ds ⊂ Dr for 0 < s < r ≤ r₀ and let B = _r>∩

0D_r. As before, we see that f(D_r\D_s) ⊂ B_r for 0 < s < r ≤ r₀, that C(f, B) = _r>∩

0f(Dr) is connected and C(f, B) ⊂Br for 0 < r≤r₀ and p(a)∈C(f, B). This implies that C(f, B) ⊂B(p(a), r) for 0< r ≤r₀, hence C(f, B) = {p(a)}, B ∩K = ∅. Since f is a light map on Rⁿ\K, we have cardB = 1. We use then the arguments from the proof of Theorem 1 in [2] to see that we can liftp from x.

We therefore proved that we can lift any pathp: [0,1]→B(0, p),p(t) = (1−t)f(x) +tyfort∈(0,1], wherex∈Rⁿ\K andy∈B(0, ρ). We take now V =B(0, ρ), x∈(Rⁿ\K)∩f⁻¹(V) and letQ be the component of f⁻¹(V) containing x. Then the map g = f|Q : Q → V is a local homeomorphism which lifts the paths and V is simply connected. This implies that g is a homeomorphism. We see thatQis the exterior of a Jordan domainDand the first half of theorem is proved.

Suppose now thatf is open and discrete on K. Let x ∈Rⁿ\K, ρ > 0 such thatf(K) ⊂B(f(x), ρ) and let U ∈ V(x) and 0 < r < ρ be fixed such that B(f(x), r) ⊂ f(U). If y ∈ S(f(x), ρ), let αy ∈ U such that f(αy) ∈ S(f(x), r)∩[f(x), y] and let Γy : [0,1]→B(f(x), ρ),Γy(t) = (1−t)f(αy) +ty for t ∈ [0,1]. Let Cρ = {y ∈ S(f(x), ρ) | Γy cannot be lifted from αy} and Γ ={pathsγy : [0, cy) →Rⁿ |y ∈ Cρ and γy is a maximal lifting of Γy from αy}. Then∞ ∈Imγfor everyγ∈Γ and by Lemma 1 we haveM_Kn−1(∞) = 0, henceM(Γ) = 0. Sincef is a map of finite distortion satisfying condition (A) onRⁿ\K, we use (ii) to see thatM(f(Γ)) = 0. This implies thatmn−1(Cρ) = 0, hence we can lift a.e. path Γ_y from α_y. Since ρ > 0 can be taken great enough, it results thatf is unbounded.

As in the first part of the proof, we can findρ >0 and a Jordan domain Dsuch that f|D:D→B(0, ρ) is a homeomorphism. Then f|∂D:∂D→ S(0, ρ) is a homeomorphism andf is open, discrete onD. Using the theorem of univalence on the border [3], we see that f is injective on D and f(D) = B(0, ρ). It results thatf :Rⁿ→Rⁿ is a homeomorpism.

Remark 1. Let f : Rⁿ → Rⁿ, f(x) = x if x ∈ B(0,1), f(x) = _|x^x_|2 if x ∈ B(0,1). Then f is conformal on Rⁿ\S(0,1), it is not an open map on S(0,1) and it is not injective. This shows that the openess of the mapf from Theorem 3 on the “singular” set K is necessary in order to ensure that f :Rⁿ→Rⁿis a homeomorphism. We also see that if n≥3, f :Rⁿ→Rⁿis a nonconstant map of finite distortion ofK(x) dilatation satisfying condition (A) and the condition

B(0,r)

exp(A(K(x)))_|x¹_|2ndx <∞, then the branch set Bf = {x ∈ Rⁿ | f is not a local homeomorphism at x} is either empty or unbounded.

Proof of Theorem 4. Since M_Kn−1(F) = 0, we have cap F = 0, hence that Rⁿ\F is a domain. Since f is nonconstant on Rⁿ\F, the functionf is open, discrete onRⁿ\F. Suppose that capf(Rⁿ\F)>0 and letK⊂Rⁿ\F be compact, connected, with cardK > 1. Then E = (f(Rⁿ\F), f(K)) is a capacitor and q(f(K)) >0, and by Lemma 2.6, page 65, in [5], there exists δ >0 such thatδ < capE. Let Γ = ∆(f(K),f(Rⁿ\F),Rⁿ) and let Γ be the family of all maximal liftings of some paths from Γ starting from some points ofK. Then Γ< f(Γ) and ifγ ∈Γ, thenγ has at least a limit point inF∪{∞}. Since

B(0,r)

exp(A(K(x)))·_|x¹_|2ndx <∞, by Lemma 1 we haveM_Kn−1(∞) = 0, henceM_Kn−1(F ∪ {∞}) = 0. This implies that M_Kn−1(Γ) = 0. We use now (i) and obtain that δ <cap(E) =M(Γ)≤M(f(Γ))≤M_Kn−1(Γ) = 0, so we reach a contradiction. We therefore proved that capf(Rⁿ\F) = 0.

Remark 2. The condition

B(0,r)

exp(A(K(x)))·_|x¹_|2ndx <∞, which en- sures thatM_Kn−1(∞) = 0, used in our generalizations of Zoric’s and Picard’s theorems, holds for instance if K(x) ≤ K_p for x ∈ B(0, p), p ≥ p₀ and

lim sup

p→∞

K_p

lnp =α < n(i.e. it holds for a locally quasiregular mapping having some logarithmic growth of the constant of quasiregularity near∞). Indeed, takeA(t) =t. If p₁≥p₀ is such that Kp < α·lnp forp≥p₁, then

B(0,p₁)

exp(A(K(x)))· 1

|x|²ⁿdx=

p≥p₁

B(0,p+1)\B(0,p)

exp(A(K(x)))· 1

|x|²ⁿdx≤

p≥p₁

B(0,p+1)\B(0,p)

exp(K_p+1)· 1

|x|²ⁿdx≤C₀·

p≥p₁

(p+ 1)^α((p+ 1)ⁿ−pⁿ)

p²ⁿ ≤

≤C₁·

p≥p₁

1

p^n−α+1 <∞. HereC₀ andC₁ are suitable constants.

We can generalize in this way Theorems 8 and 9 from [2], obtaining a generalization of Zoric’s theorem for locally quasiregular mappings having a logarithmic growth of the constant of quasiregularity near∞.

Similarly, the condition

B(b,ρ)

exp(A(K(x)))dx <∞, which ensures that M_Kn−1(b) = 0 holds for instance if K(x) ≤ Kp on B(b,¹_p) for p ≥ p₀ and lim sup

p→∞

K_p

lnp =α < n. This condition holds for locally quasiregular mappings having some logarithmic growth of the constant of quasiregularity near the critical pointb ∈ ∂D and it can be used in Theorems 1 and 4. Also, we see that the setB from Theorem 1 can be taken to be countable.

Proof of Theorem 5. Letx∈Dbe fixed and letε >0 withB(x, ε)⊂D.

We fix ε > 0 and suppose that there exist ρ > 0, r_p → 0 and f_p ∈ W such that q(fp(B(x, rp))) ≥ ρ for every p ∈ N. Then the fp are open, discrete mappings for every p ∈ N and f_p(U)∩M = ∅ for every open, nonempty U ⊂ D and every p ∈ N. Then f_p(B(x, r_p))∩M = ∅ for every p ∈ N and we can findδ >0 such that δ ≤cap (M , f_p(B(x, r_p))) for every p ∈N (see Lemma 2.6, page 65, in [15]). By (ii) we have δ ≤ cap(M , fp(B(x, rp))) ≤ cap(fp(B(x, ε)), fp(B(x, rp))) ≤cap_Kn−1(B(x, ε), B(x, rp))→ 0 ifrp → 0 and ε >0 is kept fixed, so we reach a contradiction. We therefore proved that for everyε >0 there exists δε >0 such that f(B(x, δε))⊂Bq(f(x), ε) for every f ∈W, i.e., W is equicontinuous at x.

Proof of Theorem 6. Let x ∈ D, δ > 0 be such that B(x, δ) ⊂ D and M > 0 such that |f(y)| ≤ M for every y ∈ B(x, δ) and every f ∈ W. Let W₀ =

f|B(x, δ)|f ∈W

. By Theorem 5, W₀ is equicontinuous at x, hence W is equicontinuous at x. Take on D the Euclidean metric and on Rⁿ the chordal metric. By Theorem 20.4, page 68 in [16], if (f_p)_p∈N is a sequence of mappings fromW, we can find a subsequence (f_pk)_k∈Nand a mapf :D→Rⁿ such that f_pk → f uniformly on the compact subsets of D. Ifx ∈D is fixed and Mx >0 is such that |fp_k(x)| ≤ Mx for every k ∈ N, then |f(x)| ≤ Mx, hencef takes finite values and it results thatW is a normal family.

Theorem 7. Let D⊂Rⁿbe a domain,W a family of mappings f :D→ Rⁿ of finite distortion, with the same distortion map K satisfying condition (A), and let x ∈ D be such that there exists r, δ > 0 such that B(x, δ) ⊂D and f(B(x, δ))⊂B(f(x), r) for every f ∈W. Then W is equicontinuous at x,if we take on Dand on Rⁿ the Euclidean distance.

Proof. Letf ∈W.Iff is constant onB(x, δ), thenf(B(x, ρ)) ={f(x)} ⊂ B(f(x), ε) for every 0 < ρ < δ, 0 < ε < r. Suppose that f is open, discrete on B(x, δ) and let 0 < ε < r, 0 < ρ < δ, and E = (B(x, δ), B(x, ρ)). Then f(E) = (f(B(x, δ), f(B(x, ρ)) is a capacitor and letν_nbe the function defined in [15], page 60. Keepδ >0 fixed and letρ→0 such that cap_Kn−1(E)≤ν_n(^ε_r) for 0< ρ < δ_ε. Then ν_n

|f(y)−f(x)|

r

≤ capf(E) ≤cap_Kn−1(E) ≤ ν_n(_r^ε) for every y ∈ B(x, ρ). Since νn is strictly increasing, we have |f(y)−f(x)| ≤ ε for everyy∈B(x, δε) and everyf ∈W, i.e., W is equicontinuous at x.

Remark 3. Theorem 7 shows the interesting thing that if W is a family of mappingsf :D→Rⁿof finite distortion with the same distortion mapping K and satisfying condition (A), and x is a point from D, then it is sufficient to exist a single r > 0 and a single δ >0 such that f(B(x, δ)) ⊂B(f(x), r) for every f ∈ W to obtain the equicontinuity of the family W at the point x. Taking a mappingf :D→Rⁿof finite distortion and satisfying condition (A), f_λ=f+λ, W = (f_λ)_λ∈Rn, we see immediately thatW is equicontinuous.

Since

g∈Wg(x) =Rⁿ for everyx ∈D, we cannot apply Theorem 5 to deduce the equicontinuity of the familyW, but we can use instead Theorem 7.

Theorem 8. Let D⊂Rⁿbe a domain, W a family of homeomorphisms f :D→ f(D) of finite distortion with the same distortion map K satisfying condition (A) for which there exists r >0 such that for every f ∈ W there exist af, bf ∈/ Im f with q(af, bf)≥r. Then W is equicontinuous if we take the Euclidean distance on D and the chordal distance on Rⁿ.

Proof. We follow the proof of Theorem 19.2, page 65 in [16]. Let λ_n be the function defined in [16], 12.4, page 38 and letx₀ ∈D and 0< ε < r. Let Q₀ = B(x₀, α), Q₁ = B(x₀, β) be such that 0 < α < β and B(x₀, β) ⊂ D, and let A = R(Q₀,Q₁). Then f(A) = R(f(Q₀),f(Q₁)) and q(f(Q₁)) ≥ q(af, bf) ≥ r, q(f(Q₀)) ≥ q(f(x), f(x₀)) for every x ∈ Q₀. Let x ∈ Q₀ be fixed and t = min{q(f(x₀), f(x)), r}. We keep β > 0 fixed and choose α small enough such that M_Kn−1(ΓA) ≤ λn(ε). Then λn(t) ≤ M(f(ΓA)) ≤ M_Kn−1(Γ_A)≤λ_n(ε) and sinceλ_nis increasing, we havet≤ε. Sinceε < r, we gett=q(f(x), f(x₀)). We therefore proved thatq(f(x), f(x₀))≤εfor every x∈Q₀ and every f ∈W.

Theorem 9. Let D⊂Rⁿbe a domain, W a family of homeomorphisms f :D →f(D) of finite distortion with the same dilatation map K satisfying condition (A) and such that one of the following condition is satisfied:

(i)there exist x₁, x₂∈Dand r >0 such that each f ∈W omits a point af with q(af, f(xi))≥r for i= 1,2;

(ii)there exist xi ∈D and r >0such that q(f(xi), f(xj))≥r for i=j, i, j= 1,2,3 and every f ∈W.

Then W is equicontinuous.

Proof. Suppose that condition (i) is satisfied and let D_k =D\ {x_k} for k= 1,2. By Theorem 8, the familiesW_k={f|D_k|f ∈W}are equicontinuous on Dk for k = 1,2, hence W is equicontinuous. Suppose now that condition (ii) is satisfied and letDij =D\ {xi, xj} andWij ={f|Dij|f ∈W}fori, j = 1,2,3. By Theorem 8, the families Wij are equicontinuous on Dij for i, j = 1,2,3, henceW is equicontinuous.

Corollary 1. Let D ⊂ Rⁿ be a domain, W a family of homeomor- phisms f : D → f(D) of finite distortion with the same dilatation mapK, satisfying condition (A) such that f(ai) = bi, i = 1,2,3 for every f ∈ W, where a₁, a₂, a₃ are three different points in Dand b₁, b₂, b₃ are three different points in Rⁿ. Then W is equicontinuous.

Theorem 10. Let D, D_j be domains in Rⁿ, f_j : D → D_j homeomor- phisms of finite distortion with the same dilatationmapKsatisfying condition (A) and such that f_j →f. If card(Imf) ≥3, then f :D→ D is a home- omorphism onto a domain D from Rⁿ while if f_j → f uniformly on the compact subsets from D,then f is either constant or a homeomorphism onto a domain from Rⁿ.

Proof. We follow the proof of Theorem 21.1 in [16] page 9. Letb₁, b₂, b₃ be three different points in Imf,a_k∈D such that f(a_k) =b_k fork= 1,2,3, and let r > 0 be such thatq(f(a_i), f(a_j))≥ r for i= j, i, j = 1,2,3. Then there exists j₀ ∈N such thatq(f_j(a_i), f_j(a_k))≥ ^r₂ fori=k,i, k = 1,2,3 and j≥j₀. It follows from Theorem 9, condition 2), that the familyW = (f_j)_j≥j0 is equicontinuous. By Theorem 20.3 in [16], page 68, we havefj →f uniformly on the compact subsets of D, hence f is continuous on D. By Brouwer’s theorem, it is enough to prove thatf is injective onD.

Suppose that there existsz₁, z₂ ∈D,z₁ =z₂such thatf(z₁) =f(z₂) and let r > 0 be such that z₂ ∈/ B(z₁, r). Then f_j(S(z₁, r)) separates the points f_j(z₁) and f_j(z₂), hence we can findx_j ∈S(z₁, r) such that

(1) q(fj(xj), fj(z₁))≤q(fj(z₁), fj(z₂)) for every j∈N.