Hollow quasi-Fatou components of quasiregular maps

We define a quasi-Fatou component of a quasiregular map as a connected component of the complement of the Julia set. A domain in $\mathbb{R}^d$ is called hollow if it has a bounded complementary component. We show that for each $d \geq 2$ there exists a quasiregular map of transcendental type $f: \mathbb{R}^d \to \mathbb{R}^d$ with a quasi-Fatou component which is hollow. Suppose that $U$ is a hollow quasi-Fatou component of a quasiregular map of transcendental type. We show that if $U$ is bounded, then $U$ has many properties in common with a multiply connected Fatou component of a transcendental entire function. On the other hand, we show that if $U$ is not bounded, then it is completely invariant and has no unbounded boundary components. We show that this situation occurs if $J(f)$ has an isolated point, or if $J(f)$ is not equal to the boundary of the fast escaping set. Finally, we deduce that if $J(f)$ has a bounded component, then all components of $J(f)$ are bounded.


Introduction
In the study of complex dynamics, the first example of a transcendental entire function with a multiply connected Fatou component was given by Baker [1] over half a century ago. We refer to the survey [4] for definitions and further background on complex dynamics. Since Baker's paper many authors have studied multiply connected Fatou components; see, for example, the papers [2,3,11,23,26].
In this paper we extend this study to more than two (real) dimensions, for the first time. Suppose that d 2, and that f : R d → R d is a quasiregular map of transcendental type. We refer to Section 2 for a definition of a quasiregular map; intuitively, this is a map with a bounded amount of local distortion. A quasiregular map is said to be of transcendental type if it has an essential singularity at infinity. In this setting we need a different definition of the Julia set to that used in complex dynamics. Following [6] and [10], we define the Julia set J ( f ) to be the set of all x ∈ R d such that Remark 1. If R d = R 2 , which we can identify with C, we can take f to be a transcendental entire function with a multiply connected Fatou component. Our construction in dimensions greater than two is an analogue of Baker's original construction, applied using the functions in [12].
Remark 2. When Baker constructed a multiply connected Fatou component in [1] he did not know if it was bounded. In fact, it was over ten years before he showed [2] that this is indeed the case. He later showed [3] that, in fact, this is always the case for a transcendental entire function. We do not know, however, if the hollow quasi-Fatou component of the function in Theorem 1·1 is bounded or not. It is an open question whether there exists a quasi-Fatou component that is both hollow and unbounded.
We prove two results, which concern the cases that a hollow quasi-Fatou component is either bounded or unbounded. The first shows that quasi-Fatou components of quasiregular maps of transcendental type that are both bounded and hollow have properties very similar to those of multiply connected Fatou components of transcendental entire functions. In order to state this result we need to give a number of definitions, all of which are familiar from complex dynamics.
If U 0 is a quasi-Fatou component of f , then we let U k be the quasi-Fatou component of f containing f k (U 0 ), for k ∈ N. (Note that, even for entire f , we cannot assume that U k = f k (U 0 ), although this equality can hold; see Lemma 5·1 below and [19, lemma 4·1]). If U n U m , for n m, then we say that U 0 is wandering.
The escaping set was first studied for a general transcendental entire function in [13], and plays an important role in complex dynamics. This set was first studied for a quasiregular map in [8].
We also use two subsets of the escaping set which themselves now have an important role in complex dynamics. The fast escaping set A( f ) can be defined by Here M(R, f ) denotes the maximum modulus function with respect to the first variable, and R > 0 can be taken to be any value such that M k (R, f ) → ∞ as k → ∞. For a transcendental entire function this form of the definition of A( f ) was first used in [21]. This definition was first used for a quasiregular map of transcendental type in [7], where it was shown to be independent of R, and equivalent to two other definitions. We also define the set It is known that for a transcendental entire function f , the sets A R ( f ), A( f ) and I ( f ) can have a structure known as a spider's web. We refer to [21,22] for more background on this structure, which is defined as follows.
Definition 1·2. A set E ⊂ R d is a spider's web if E is connected, and there exists a sequence (G n ) n∈N of bounded full domains with G n ⊂ G n+1 , for n ∈ N, ∂G n ⊂ E, for n ∈ N, and n∈N G n = R d .
We are now able to state our result. Here we denote the Euclidean distance from a point x to a set U ⊂ R d by dist(x, U ) = inf y∈U |x − y|. We say that a set U ⊂ R d surrounds a set V ⊂ R d if V is contained in a bounded component of the complement of U . THEOREM 1·3. Suppose that f : R d → R d is a quasiregular function of transcendental type. Suppose that U 0 is a quasi-Fatou component of f which is bounded and hollow. Let R > 0 be such that M k (R, f ) → ∞ as k → ∞. Then: (a) each U k is bounded and hollow, U k+1 surrounds U k for all large k, and also Remark 3. If f is a transcendental entire function and U 0 is a Fatou component of f , then the assumption that U 0 is bounded is not required. In this case part (a) is a result of Baker [3, theorem 3·1], and part (b) and part (c) are due to Rippon  The following corollary of Theorem 1·3 and Theorem 1·4 is immediate, and is a generalisation of a well-known fact in complex dynamics.
Finally we consider the implications of two possible configurations of the Julia set of a quasiregular map, f , of transcendental type. Firstly, it is known [9, theorem 1·1] that Secondly, it is known that for many quasiregular maps of transcendental type, J ( f ) is perfect. The paper [10] shows that this holds in a variety of circumstances. It is not known, however, if it is ever possible for J ( f ) to have an isolated point.
The following theorem considers the case that either

has an isolated point. Then f has a quasi-Fatou component which is unbounded and hollow.
A key tool in the proof of these results is the following theorem, which may be of independent interest. Here we denote by B(a, r ) the open ball of Euclidean radius r , centred at a point a ∈ R d .
Then there exists ∈ N such that Note that the hypothesis (1·4) is satisfied if G ∂ A( f ) 6. We give this slightly more general result in view of potential applications.
The proof of Theorem 1·7 uses a certain conformally invariant metric. Very roughly, the theorem holds since f cannot increase this metric too much, whereas the points x and y must iterate far apart in the Euclidean metric. This can be used to show that the boundary of the topological hull of f k (G) must be far from these iterates, for large values of k.
The subsequent article [19] contains further results about hollow quasi-Fatou components of quasiregular maps.
The structure of this paper is as follows. First, in Section 2 we recall the definitions of quasiregularity, capacity and the modulus of a curve family, and we give some known results required in the rest of the paper. Next, in Section 3 we prove Theorem 1·1. In Section 4 we prove Theorem 1·7. Finally in Section 5 we prove Theorem 1·3, Theorem 1·4 and Theorem 1·6.

Quasiregular maps, capacity and the modulus of a curve family
We refer to the monographs [20] and [25] for a detailed treatment of quasiregular maps.
Here we only recall the definition and the main properties used. If consists of the functions f : G → R d for which all first order weak partial derivatives exist and are locally in where D f (x) denotes the derivative, denotes the norm of the derivative, and J f (x) denotes the Jacobian determinant. We also let The condition that (2·1) holds for some K O 1 is equivalent to the condition that for some K I 1. The smallest constants K O and K I such that (2·1) and (2·2) hold are called the outer and inner dilatation of f and are denoted by The dilatation of f is denoted by We say that f is K -quasiregular if K ( f ) K , for some K 1.
If f and g are quasiregular, with f defined in the range of g, then f • g is quasiregular and [20, theorem II·6·8] Many properties of holomorphic functions extend to quasiregular maps. In particular, we frequently use the fact that non-constant quasiregular maps are open and discrete.
We need a result on the growth of the maximum modulus of a quasiregular map of transcendental type; see [5, An important tool in the study of quasiregular mappings is the capacity of a condenser, and we recall this concept briefly. Suppose that A ⊂ R d is open, and C ⊂ A is non-empty and compact. The pair (A, C) is called a condenser, and its capacity, which we denote by cap (A, C), is defined by where the infimum is taken over all non-negative functions u ∈ C ∞ 0 (A) which satisfy u(x) 1, for x ∈ C.
If If C is closed and cap C = 0, then C is, in a sense, a small set. In particular we use the fact [20, VII·1·15] that the Hausdorff dimension of C is zero. In particular, if C has an interior point, then cap C > 0.
A second important tool in the study of quasiregular maps is the concept of the modulus of a curve family; we refer to [20, chapter II] and [25, chapter 2] for a detailed discussion. Suppose that is a family of paths in R d . A non-negative Borel function ρ : R d → R {∞} is called admissible if γ ρ ds 1, for all locally rectifiable paths γ ∈ . We let F( ) be the family of all admissible Borel functions, and let the modulus of be given by Suppose that G ⊂ R d is a domain, and E, F are subsets of G. We denote by (E, F; G) the family of all paths which have one endpoint in E, one endpoint in F, and which otherwise are in G.
A connection between the capacity of a condenser and the modulus of a path family is the fact [20, proposition II·10·2] that if E is a compact subset of G, then cap(G, E) = M( (E, ∂G; G)).
Next we introduce a conformal invariant which is a useful alternative to the hyperbolic metric when working with quasiregular maps. Let G ⊂ R d be a domain, and define a function μ G by where the infimum is taken over curves C xy which are contained in G and join x and y; see [25, p. 103] for this definition and more background. It is known that μ G (x, y) is a conformal invariant, and is a metric if cap ∂G > 0. It is noted in [25] that if D ⊂ G is a domain, then The "transformation formula" for μ G is as follows [25, theorem 10·18]. G (a, b), for a,

Proof of Theorem 1·1
We construct our function using a theorem of Drasin and Sastry in [12]. We need to introduce some terminology before we can state the result we use from that paper.
Suppose that ν : R → R is continuous, positive and increasing, and that ν(r ) → ∞ as r → ∞. Let n 0 be an integer greater than ν(0), and define a sequence of integers (r n ) n n 0 by r n = max{r : ν(r ) = n}. (3·1) In [12] it is assumed that In fact (3·2) is only used in [12] to deduce that n log r n+1 r n −→ ∞ as n −→ ∞.
In our case it is somewhat easier to check (3·3) directly, and so we use this equation in the statement of Drasin and Sastry's theorem. It is helpful to write down a formula for an exceptional set. For ∈ (0, 1), we define a union of closed intervals

Define a function
Finally, it is helpful to work with the maximum norm, which is defined by A statement of Drasin and Sastry's result, which includes part of their construction, is as follows.
THEOREM 3·1. Suppose that d 2, that ν(r ) and L(r ) are as above, and that (3·3) is satisfied. Then there exist a quasiregular map of transcendental type f : R d → R d , and constants c, C, R 0 > 0 and ∈ (0, 1) such that The form of the exceptional set E in the statement of Theorem 3·1, which is not made explicit in [12], can be obtained as follows. First, we can deduce from [12, equation (2·10)], together with the paragraph preceding it, that there exists ∈ (0, 1) such that if r n+1 > r n , then the interval [r n , r n+1 ] lies in the interval labelled J 0 n by Drasin and Sastry, for all sufficiently large values of n. Second, the fact that (3·5) holds when ||x|| ∞ ∈ J 0 n follows from [12, equation (3·6)].
Proof of Theorem 1·1. Roughly speaking, we construct a quasiregular function f of transcendental type with the following properties: (a) the function f behaves like a power map in very large "square rings"; (b) subrings of these can be defined in such a way that f maps these subrings into each other. Note that this is, essentially, the same idea as Baker's original construction of a multiply connected Fatou component of a transcendental entire function using an infinite product. Properties (a) and (b) are achieved iteratively: if r n is defined, then ν and r n+1 are chosen so that r n+1 is much larger than r n (to satisfy (a)), and so that if ||x|| ∞ = r n , then || f (x)|| ∞ is approximately equal to r n+1 (which leads to (b)). Property (b) then ensures that these subrings lie in Q F( f ), and the theorem follows.
We now give the full detail of the proof. Let d 2 be an integer. We first construct a function ν with certain properties. We then invoke Theorem 3·1 to obtain a quasiregular map f : R d → R d such that (3·5) is satisfied. Finally we prove that this function has a hollow quasi-Fatou component.
The construction of ν is as follows. First choose an integer n 0 2 and a real number R > 4. Set ν(r ) = n 0 , for 0 r R . Since we shall ensure that ν(r ) > n 0 , for r > R , this implies that r n 0 = R .
We complete the definition of ν, and the sequence (r n ) n>n 0 iteratively. Suppose that n n 0 , that r n has been defined, and that ν(r ) has been defined for 0 r r n . We set and let ν(r ) be linear; in other words, we set ν(r ) = r − r n r n+1 − r n + n, for r n r r n+1 .
Note that if n n 0 , then It follows from the choices of n 0 and R that ν is positive, continuous and increasing, tends to infinity, and that (3·3) holds. Hence we can let f : R d → R d and constants c, C, R 0 > 0 and ∈ (0, 1) be as in Theorem 3·1.
We complete the proof of the theorem by showing that f has a hollow quasi-Fatou component. For each sufficiently large integer n n 0 , define a "square ring" A n = {x : 2r n < ||x|| ∞ < r n+1 } .
We claim that for all sufficiently large values of n, we have that f (A n ) ⊂ A n+1 . It then follows from (1·1) that A n ⊂ Q F( f ). It is easy to see that the theorem follows from this fact, since the Julia set of f is not empty.
Choose N 0 n 0 sufficiently large that, for n N 0 we have that 2r n < r n+1 , that c2 n 2, and that C n . Suppose that n N 0 , and that x ∈ A n . Then, by (3·5), (3·6), (3·7) and since L(r ) is increasing For the same reasons, Hence f (x) ∈ A n+1 , and this completes the proof. Suppose that f : R d → R d is a quasiregular function of transcendental type, and that G is a domain. We can assume we have T ( f k (G)) R d , for k ∈ N, as otherwise there is nothing to prove. Let x, y ∈ G, r > 0 and 0 , k 0 ∈ N be as in the statement of the theorem. By Lemma 2·1 and (1·4), we can assume that 2| f k (y)| | f k (x)|, for all sufficiently large k. (4·1) We claim that there exists k 2 ∈ N such that Since | f k+ 0 (y)| M k−1 (r, f ), for all k k 0 , it is easy to see that this claim implies that from which Theorem 1·7 follows with = k 2 + 0 .
Since G k is full, we deduce that (5·2) Let R > 0 be sufficiently large that M k (R, f ) → ∞ as k → ∞. Since we have that G 0 J ( f ) 6, it follows by (1·3) that G 0 ∂ A( f ) 6, and so we can apply Theorem 1·7 with G = G 0 . We deduce, by Theorem 1·7 and (5·2) that there exists ∈ N such that Theorem 1·3 (a) and (b) follow.
To prove part (c), we note first that it follows from Theorem 1·3 (b) and (1·3) that the complement of A R ( f ) has a bounded component. The fact that the sets A R ( f ) and A( f ) are both spiders' webs follows immediately from [7, proposition 6·2].
It remains to show that I ( f ) is a spider's web. Since A( f ) is a spider's web, and so connected, we can let I 0 be the component of I ( f ) which contains A( f ). By definition I 0 is a spider's web. We show that, in fact, I ( f ) = I 0 .
First, suppose that x ∈ I ( f ) J ( f ). Then x ∈ ∂ A( f ), by (1·3), and so x ∈ I 0 . Thus I 0 {x} is connected and, since x ∈ I ( f ) and I 0 is a component of I ( f ), it follows that x ∈ I 0 . Thus (5·3) Next, suppose that x ∈ I ( f ) Q F( f ). Let V 0 be the component of Q F( f ) containing x. We claim that V 0 ⊂ I ( f ). It is easy to see that the fact that I ( f ) = I 0 follows from this claim, by (5·3).
Recall that V n (resp. U n ) denotes the Fatou component containing f n (V 0 ) (resp. f n (U 0 )). If V n = U m , for some n, m ∈ N, then V n ⊂ A( f ), by Theorem 1·3 (b). So we can assume that V n U m , for n, m ∈ N. (5·4) For all sufficiently large values of n, let B n denote the intersection of the unbounded complementary component of U n with T (U n+1 )\U n+1 . Since f n (x) → ∞, it follows by (5·4) that there is a sequence (k n ) n∈N such that f n (x) ∈ B k n , for all sufficiently large n ∈ N, and k n → ∞ as n → ∞.
It follows by (5·4) that V n ⊂ B k n , for all sufficiently large n ∈ N, and the result follows.
Proof of Theorem 1·4. Suppose that f : R d → R d is a quasiregular function of transcendental type. Suppose that U is a quasi-Fatou component of f which is unbounded and hollow.
Let E be a bounded complementary component of U . We claim that there is a bounded full domain, G, such that E ⊂ G and ∂G ⊂ U . We prove this claim as follows. For each n ∈ N, let V n be the component of {y : dist(y, U c ) 1/n} that contains E.
Suppose that all the sets V n are unbounded. Let ρ > 0 be sufficiently large that E ⊂ B(0, ρ/2). For each n ∈ N, let V * n be the component of V n B(0, ρ) containing E. It is easy to see that each V * n is a continuum. Let V * = n∈N V * n . Then V * is a nested intersection of continua, and so [17, theorem 1·8] is a continuum. In particular V * is connected.