TRANSCENDENTAL ENTIRE FUNCTIONS WHOSE JULIA SETS CONTAIN ANY INFINITE OF QUASICONFORMAL OF QUADRATIC JULIA

. We prove that for any inﬁnite collection of quadratic Julia sets, there exists a transcendental entire function whose Julia set contains quasicon- formal copies of the given quadratic Julia sets. In order to prove the result, we construct a quasiregular map with required dynamics and employ the qua- siconformal surgery to obtain the desired transcendental entire function. In addition, the transcendental entire function has order zero.


1.
Introduction. Let f : C → C be an entire function. The Fatou set F (f ) is the set of normality in the sense of Montel for the family {f n } ∞ n=1 or F (f ) = z ∈ C : {f n } ∞ n=1 is a normal family in some neighbourhood of z , where f n = f • · · · • f is n iterates of f . By definition, the Fatou set is open. The escaping set I(f ) is the set of points that escape to infinity under iteration, I(f ) = z ∈ C : f n (z) → ∞ as n → ∞ .
The Julia set J (f ) is the complement of the Fatou set, J (f ) = C \ F (f ). The Julia set is closed and is equal to the topological boundary of the escaping set. Moreover, the Julia set is equal to the closure of the set of repelling periodic points. If f is a polynomial of degree at least two, the complement of the escaping set is called the filled Julia set K(f ) or K(f ) = z ∈ C : {f n (z)} ∞ n=1 is bounded . In this case, J (f ) = ∂K(f ) holds and the escaping set is a component of the Fatou set. However, if f is a transcendental entire function, then I(f ) ∩ J (f ) = ∅. These four sets F (f ), K(f ), J (f ) and I(f ) are completely invariant under f , namely f (F (f )) ⊂ F (f ) and f −1 (F (f )) ⊂ F (f ) and so on. For basic properties of these sets, we refer to [4,11].
The purpose of this paper is to construct a transcendental entire function whose Julia set contains quasiconformal copies of infinitely many quadratic Julia sets, or Julia sets of quadratic polynomials. In other words, for any sequence {c j } ∞ j=1 of complex numbers, we construct a transcendental entire function f such that there are infinitely many quadratic-like maps (f, U j , V j ) and each (f, U j , V j ) is hybrid equivalent to the quadratic polynomial p cj for j ≥ 1, where p c (z) = z 2 + c. For details about quadratic-like maps and hybrid equivalence, see Definition 2.2 and Theorem 2.3. The Julia set J (f ) of the transcendental entire function f we construct contains a quasiconformal copy of the Julia set J (p cj ) of the quadratic polynomial p cj , or the image of J (p cj ) under a quasiconformal map. In [9], the author constructed a transcendental entire function whose Julia set contains quasiconformal copies of finitely many quadratic Julia sets. It is a finite version of our purpose. Theorem 1.1 ([9, Theorem B]). Let n ≥ 2 be an integer. For any c 1 , c 2 , . . . , c n ∈ C, there exist a transcendental entire function f and distinct bounded simply connected domains U 1 , U 2 , . . . , U n and V satisfying U j ⋐ V for j = 1, 2, . . . , n such that each (f, U j , V ) is a quadratic-like map and is hybrid equivalent to the quadratic polynomial p cj for j = 1, 2, . . . , n.
The main result of this paper is a generalization of Theorem 1.1.
Theorem A. For any sequence {c j } ∞ j=1 of complex numbers, there exist a transcendental entire function f and two sequences of distinct bounded simply connected is a quadratic-like map and is hybrid equivalent to the quadratic polynomial p cj for j ≥ 1.
The idea of the proof of Theorem A is the following. First, we interpolate polynomials to construct a quasiregular map which is a map with required dynamics. Next, we cut off countably many small disks and interpolate the quasiregular map and an infinite collection of quadratic polynomials in the small disks. Finally, we employ the quasiconformal surgery for the constructed quasiregular map to obtain a transcendental entire function with the desired property. While the method of construction in this paper is similar to the one in the author's previous paper [9], we can not use the same strategy in [9] to construct a transcendental entire function whose Julia set contains quasiconformal copies of infinitely many quadratic Julia sets. Theorem A is the first example of a transcendental entire function whose Julia set with quasiconformal copies of infinitely many quadratic Julia sets. In addition, we can calculate the order of the transcendental entire function.
Theorem B. The transcendental entire function f obtained by Theorem A has order zero.

2.
Background. In this section, we review some basic definitions and results. For more details, we refer to [1,6,8,11,12,15]. Definition 2.1 (Quasiconformal maps). Let U be a domain in C and let ϕ : U → ϕ(U ) be a sense-preserving C 1 homeomorphism. We define the complex dilatation of ϕ as µ ϕ = ϕz ϕ z and the dilatation of ϕ as We say that ϕ is a K-quasiconformal map, Note that a C 1 homeomorphism is conformal if and only if it is 1-quasiconformal.
Definition 2.2 (Polynomial-like maps). The triple (g, U, V ), consisting of bounded simply connected domains U and V such that U ⋐ V and a holomorphic proper map g : U → V of degree d, is called a polynomial-like map of degree d. The filled Julia set K(g, U, V ) of a polynomial-like map (g, U, V ) is defined as K(g, U, V ) = z ∈ U : g n (z) ∈ U for all n ≥ 0 and the Julia set J (g, U, V ) as J (g, U, V ) = ∂K(g, U, V ). In the case that d = 2, the triple (g, U, V ) is called a quadratic-like map.
Theorem 2.3 (Straightening Theorem [7]). Every polynomial-like map is hybrid equivalent to a polynomial of the same degree. Namely, for any polynomial-like map If K(g) is connected, p is unique up to conjugation by an affine map.
Now, we define quasiregular mappings and state important results concerning them. Throughout this paper, "log" denotes the principal branch of the logarithm.
Definition 2.4 (Quasiregular maps). Let U be a domain in R n with n ≥ 2. A mapping g : U → R n is quasiregular if g belongs to the Sobolev space W 1 n,loc (U ) and there exists K ≥ 1 such that almost everywhere in U , where ||Dg(x)|| is the operator norm of the derivative and J g (x) is the Jacobian determinant of g at x ∈ U . The smallest K for which the above inequality holds is the outer dilatation K O (g) of g. If g is quasiregular, then we have J g (x) ≤ K ′ ℓ(Dg(x)) n almost everywhere in U for some K ′ ≥ 1, where ℓ(Dg(x)) = inf |v|=1 |Dg(x)v|. The smallest K ′ for which the above inequality holds is the inner dilatation K I (g) of g. The maximal dilatation of g is K(g) = max{K O (g), K I (g)} and a quasiregular mapping is called K-quasiregular if K(g) ≤ K.
Note that non-constant quasiregular maps are open and discrete. The relationships K O (g) ≤ K I (g) n−1 and K I (g) ≤ K O (g) n−1 hold as is seen by linear algebra. Therefore K O (g) = K I (g) for n = 2. A quasiregular homeomorphism is called quasiconformal. By Stoïlow's theorem, a quasiregular mapping g : is analytic. For more information on quasiregular mappings, we refer to [14]. Lemma 2.5 (Interpolation [10, Lemma 6.2]). Let k ∈ N, 0 < r 1 < r 2 and ϕ j (z) be analytic on a neighborhood of |z| = r j such that ϕ j | |z|=rj goes around the origin k-times (j = 1, 2). If log ϕ 2 r 2 e iy r k and hold for every y ∈ [0, 2π] and for some positive constants δ 0 and δ 1 satisfying then there exists a quasiregular map without critical points such that H = ϕ j on |z| = r j (j = 1, 2) and satisfies Theorem 2.6 (Quasiconformal surgery [10, Theorem 3.1]). Let g : C → C be a quasiregular mapping. Suppose that there are disjoint measurable sets E j ⊂ C (j = 1, 2, . . . ) satisfying: (1) for almost every z ∈ C, the g-orbit of z passes E j at most once for every j; 3. Key Lemma. The following lemma plays an important role to construct a quasiregular map with required dynamics, which is a seed of a desired transcendental entire function. The key Lemma is an analog of the one introduced by Kisaka and Shishikura in [10].

Proof of Theorem A.
In this section, we prove Theorem A. Let Ann(c ; r 1 , r 2 ) be the open annulus centered at c of inner radius r 1 and outer radius r 2 and let Ann(r 1 , r 2 ) = Ann(0 ; r 1 , r 2 ). We first construct a quasiregular map g : C → C with required dynamics by modifying Bergweiler's method [5], Osborne's one [13] or the author's one [9], which are based on the ideas of Kisaka and Shishikura [10]. The construction uses the sequences (R m ), We define the quasiregular map g as ψ m−1 on Ann(T m−1 , P m ), η m on Ann(Q m , S m ) and ψ m on Ann(T m , P m+1 ). Next, we interpolate the maps ψ m−1 and η m in the annulus Ann(P m , Q m ), and the maps η m and ψ m in the annulus Ann(S m , T m ) respectively. Moreover, we alter the definition of g in the small disk |z − R m(j) | ≤ r m(j) in Ann(Q m(j) , S m(j) ) to a quadratic polynomial which is conjugate to p cj , where m(j) is a positive integer depending on the complex number c j given in Theorem A. Furthermore, we interpolate the quadratic polynomial defined in the small disk |z − R m(j) | ≤ r m(j) and η m(j) in the small annulus Ann(R m(j) ; r m(j) , ρ m(j) ) in Ann(Q m(j) , S m(j) ). Finally, we employ the quasiconformal surgery for the constructed quasiregular map to obtain a transcendental entire function with the desired property.
Taking R 1 > e gives γ > 1 and thus If R 1 and hence γ is sufficiently large, we obtain that for all m ≥ 1. Therefore, the inequality holds for all m ≥ 1. Further, we define sequences (a m ), (b m ), (r m ), (ρ m ) and (A m ) as By definition of (R m ) and (a m ), ψ m maps Ann(R m , R m+1 ) onto Ann(R m+1 , R m+2 ). The finite critical points of η m are 0, R m and mR m /(m + 1). The origin is a superattracting fixed point of η m . It is easy to see that Notation (Sequences with index j). Let the sequence {c j } ∞ j=1 ⊂ C be in Theorem A, p c (z) = z 2 + c and q c,A (z) = p c (Az)/A = Az 2 + c/A, where c ∈ C and A ∈ C \ {0} are constants.
Lemma 4.1. Suppose that γ is sufficiently large. For the sequence {c j } ∞ j=1 in Theorem A, there exists a subsequence (m(j)) ∞ j=1 of (m) ∞ m=1 with m(j+1) > m(j) ≥ 2 for any j ≥ 1 such that the maps h m(j) and q cj ,A m(j) can be interpolated on r m(j) ≤ |z| ≤ ρ m(j) with a quasiregular map I j by Lemma 2.5 and I j satisfies Proof. If γ is large enough, the origin is the only critical point of h m inside |z| = ρ m , and by Rouché's theorem, h m has two zeros inside |z| = ρ m for all m ≥ 1. A direct calculation shows that there exists a subsequence (m(j)) ∞ j=1 of (m) ∞ m=1 with m(j + 1) > m(j) ≥ 2 such that q cj ,A m(j) has two zeros inside |z| = r m(j) . Hence, we apply Lemma 2.5 by taking and changing k to 2. First, we check the inequality (2.1) in Lemma 2.5. For Then the following holds as m tends to infinity: where Therefore, we obtain that as m tends to infinity. Hence, for j ≥ 1, there exists a subsequence (m(j)) ∞ j=1 of (m(j)) ∞ j=1 , which we relabel as m(j), such that Next, we check the inequality (2.2) in Lemma 2.5. Since (4.2), if m tends to infinity, holds. Moreover, we obtain that where Since B m → 1, Ω m → ∞ and s m → 0 as m tends to infinity, then Hence, for j ≥ 1, there exists a subsequence (m(j)) ∞ j=1 of (m(j)) ∞ j=1 , which we relabel as m(j), such that the inequality (4.4), hold. Finally, we check the inequality (2.3) in Lemma 2.5. If γ is sufficiently large, then the inequality holds for any j ≥ 1. By Lemma 2.5, then there exists a quasiregular map without critical points such that I j = q cj ,A m(j) on |z| = r m(j) and I j = h m(j) on |z| = ρ m(j) and satisfies Construction of a quasiregular map with required dynamics. Let m ≥ 2 an integer. First, we interpolate ψ m−1 and η m for m ≥ 2. We consider the functions We apply Lemma 3.1 by taking By definition, Then |ω| ≥ 2ρ ♭ and ρ ♭ ≥ eλ ♭ because we obtain that for all m ≥ 1 if γ is large enough. Therefore, by part (2) of Lemma 3.1, there exists a quasiregular map Hence, we define the map g on Ann(T m−1 , S m ) as follows: Similarly, we interpolate η m and ψ m for m ≥ 1. We consider the functions We apply Lemma 3.1 by taking By definition, Then ρ ♯ ≥ 2|ω| and λ ♯ ≥ eρ ♯ because Since |log τ | = log (m + 1) 2 · m + 1 m + 2 · 1 + 1 m 2m ≤ log (m + 1) 2 e 2 = 2 {log(m + 1) + 1} , we obtain that for all m ≥ 1 if γ is large enough. Therefore, by part (1) of Lemma 3.1, there exists a quasiregular map without critical points such that g ♯ m (z) = η ♯ m (z) = τ m z 2m (z − 1) 2 on |z| = ρ ♯ and g ♯ m (z) = ψ ♯ m (z) = z 2m+2 on |z| = λ ♯ with

KOH KATAGATA
Hence, we define the map g on Ann(Q m , P m+1 ) as follows: Moreover, we define the map g as η 1 on |z| ≤ Q 1 . Therefore, the map g is K mquasiregular on E m for m ≥ 1, where Finally, for z ∈ Ann(Q m(j) , S m(j) ), we redefine the map g as Then, the map g is K ′ j -quasiregular on D j for j ≥ 1, where D j = Ann R m(j) ; r m(j) , ρ m(j) and K ′ j = 1 + 1 j 2 .
Note that the point m(j)R m(j) /(m(j) + 1), a critical point of η m(j) , does not belong to the disk |z − R m(j) | ≤ ρ m(j) for all j ≥ 1..
Proof. First, we show that |g(z)| ≤ Q m(j)+1 on D j for j ≥ 1. By the maximum principle, for z ∈ D j , the inequality holds for all j ≥ 1 if γ is large enough. Next, we show that |g(z)| ≥ P m(j)+1 on D j for j ≥ 1. Since g(z) has no zeros in D j , by the minimum principle, for z ∈ D j , the inequality holds. Then, there exists a subsequence (m(j)) ∞ j=1 of (m(j)) ∞ j=1 , which we again relabel as m(j), such that the inequalities and e 2 √ γ·(2m(j)+2)!!−γ·(2m(j))!! < 2 3 hold for all j ≥ 1. Therefore, the inequality holds for all j ≥ 1.
Proof. Note that g(z) has no critical points in Ann(S m , Q m+1 ) and First, we show that |g(z)| ≤ Q m+2 on Ann(S m , Q m+1 ) for m ≥ 1. By the maximum principle, for z ∈ Ann(S m , Q m+1 ), the inequality holds for all m ≥ 1 if γ is large enough. Next, we show that |g(z)| ≥ S m+1 on Ann(S m , Q m+1 ) for m ≥ 1. Since g(z) has no zeros in Ann(S m , Q m+1 ), by the minimum principle, for z ∈ Ann(S m , Q m+1 ), the inequality holds for all m ≥ 1. Proof of Theorem A. Note that the map g is defined as We will check that disjoint measurable sets E m (m = 1, 2, . . . ) and D j (j = 1, 2, . . . ) defined as E m = Ann(S m , T m ) ∪ Ann(P m+1 , Q m+1 ) and D j = Ann R m(j) ; r m(j) , ρ m(j) satisfy the assumption of Theorem 2.6. By Lemma 4.2 and Lemma 4.3, for every z ∈ C, the g-orbit of z passes E m and D j at most once for every m and j. By construction, the map g is K m -quasiregular on E m and K ′ j -quasiregular on D j , where K m = 1 + 1/m 2 and K ′ is finite and K ∞ = (sinh π/π) 2 . Furthermore, the quasiregular map g is holomorphic almost everywhere outside ∞ m=1 E m ∪ ∞ j=1 D j . Therefore, by Theorem 2.6, there exists a K ∞ -quasiconformal map ϕ : C → C such that f = ϕ • g • ϕ −1 is a transcendental entire function. We normalize ϕ as ϕ(0) = 0 and ϕ(1) = 1. Let and let U j be the connected component of f −1 (V j ) containing ϕ(R m(j) ). Since g(D j ) ⊂ Ann(P m(j)+1 , Q m(j)+1 ) by Lemma 4.2, U j is contained by the bounded component of C \ ϕ(D j ). Therefore, we obtain that U j ⋐ V j . Moreover, U j contains only one critical point ϕ(R m(j) ) because the critical point m(j)R m(j) /(m(j) + 1) of g does not belong to the disk |z − R m(j) | < ρ m(j) . Then each triple (f | Uj , U j , V j ) becomes a quadratic-like map. By the straightening theorem, the quadratic-like map (f | Uj , U j , V j ) is hybrid equivalent to the quadratic polynomial q cj ,A m(j) , which is affine conjugate to the quadratic polynomial p cj : z → z 2 + c j for all j ≥ 1. The proof of Theorem A is completed. 5. The order of the entire function obtained by Theorem A. We can calculate the order of the entire function f = ϕ • g • ϕ −1 obtained by Theorem A, considering the dynamics of the quasiregular map g. where M (r, f ) = max |z|=r |f (z)| is the maximum modulus function. The order of a quasiregular map can be defined in the same way.
For an ordered quadruple a, b, c, d of distinct points in R n , we define the absolute cross-ratio | a, b, c, d | as where q is the spherical metric in R n , namely x, y ∈ R n 1 1 + |x| 2 if x ∈ R n and y = ∞.
A Möbius map of R n onto itself preserves the absolute cross-ratio. The absolute cross-ratio depends on the order of the points. Note that | 0, e 1 , x, ∞ | = |x|. The function λ(K) is a distortion function of the theory of plane quasiconformal mappings. For details, refer to [2]. Anderson, Vamanamurthy and Vuorinen [3] proved that the inequality e π(K−1) < λ(K) < e a(K−1) holds, where 1 < K < ∞ and a = 4.37688.