Quartic Julia sets including any two copies of quadratic Julia sets

. If the Julia set of a quartic polynomial with certain conditions is neither connected nor totally disconnected, there exists a homeomorphism be- tween the set of all components of the ﬁlled-in Julia set and some subset of the corresponding symbol space. The question is to determine the quartic polyno- mials exhibiting such a dynamics and describe the topology of the connected components of their ﬁlled-in Julia sets. In this paper, we answer the question, namely we show that for any two quadratic Julia sets, there exists a quartic polynomial whose Julia set includes copies of the two quadratic Julia sets.

If some critical orbits but not all critical orbits diverge, then the Julia set is disconnected and not generally totally disconnected. The author [3] simplified the dynamics of some quartic polynomial on its filled-in Julia set when the Julia set is disconnected and not totally disconnected.
Let f be a quartic polynomial and G = G f the Green's function associated with the filled-in Julia set K(f ). For a polynomial p of degree d, the Green's function G p is defined as G p (z) = lim n→∞ 1 d n log + p n (z) , where log + x = max{ log x, 0 }. The Green's function G(z) is zero for z ∈ K(f ) and positive for z ∈ I(f ). By definition, the identity G(f (z)) = 4 G(z) holds. The locus G −1 (x) with x > 0 is called an equipotential curve around the filled-in Julia set K(f ). Note that f maps the equipotential curve G −1 (x) to the equipotential curve G −1 (4x) by a four-to-one fold covering map. Let ω 1 , ω 2 and ω 3 be different finite critical point of f . We assume that G(ω 1 ) = G(ω 2 ) = 0 and G(ω 3 ) > 0. The assumption indicates that ω 1 , ω 2 ∈ K(f ) and ω 3 ∈ I(f ). Definition 1.2 (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-in Julia set K(g) of a polynomial-like map (g, U, V ) is defined as K(g) = z ∈ U : g n (z) ∈ U for all n ≥ 0 and the Julia set J(g) as J(g) = ∂K(g). In the case that d = 2, the triple (g, U, V ) is called a quadratic-like map. Theorem 1.3 (Straightening Theorem [2,5]). Every polynomial-like map is hybrid equivalent to a polynomial of the same degree. Namely, for any polynomial-like map (g, U, V ) of degree d ≥ 2, there exist a polynomial p of degree d, a neighborhood W of K(g) in U and a quasiconformal map ϕ : W → ϕ(W ) such that (1) ϕ (K(g)) = K(p), (2) the complex dilatation µ ϕ of ϕ is zero almost everywhere on K(g), is connected, p is unique up to conjugation by an affine map.
We assume that the A-B kneading sequence of ω 1 is (AAA · · · ) and the A-B kneading sequence of ω 2 is (BBB · · · ). This implies that filled-in Julia sets K(f | U A ) and K(f | U B ) are connected. Let Σ 6 = {1, 2, 3, 4, A, B} N be the symbol space on 6 symbols. We define its subset Σ in Theorem 1.4 as follows : s = (s n ) ∈ Σ if and only if (1) Figure 2. Equipotential curves. The red one is G −1 (G(0)). It is a topological figure-eight through the origin. The blue one is G −1 (G(f (0))). It is a topological circle through the critical value f (0) = 14/9.   Figure 4. Equipotential curves. The red one is G −1 (G(0)) and the blue one is G −1 (G(f (0))).
filled-in Julia set K(f | U B ). These backward components converge to a repelling fixed point on ∂K(f | U A ).
. Let f be a quartic polynomial. Suppose that (a) its finite critical points ω 1 , ω 2 ∈ K(f ) and ω 3 ∈ I(f ) are all different, (b) the Julia set J(f ) is disconnected but not totally disconnected, (c) the A-B kneading sequence of ω 1 is (AAA · · · ) and the A-B kneading sequence of ω 2 is (BBB · · · ). Then there exist a subset Σ of the symbol space Σ 6 and a homeomorphism Λ : The black region is the set of all a such that the critical The question is to determine the quartic polynomials exhibiting the above dynamics and describe the topology of the connected components of their filled-in Julia sets. In this study, we construct the quartic polynomial satisfying the assumptions of Theorem 1.4. The answer is the following theorem.
Theorem A. For any two quadratic Julia sets, there exists a quartic polynomial whose Julia set includes copies of the two quadratic Julia sets. More precisely, for any c 1 , c 2 ∈ C, there exist a quartic polynomial f and distinct three bounded simply connected domains U 1 , U 2 and V satisfying U 1 V and U 2 V such that (f, U 1 , V ) and (f, U 2 , V ) are quadratic-like maps and they are hybrid equivalent to the quadratic polynomials p c1 and p c2 respectively, where p c (z) = z 2 + c.
Corollary B. If c 1 and c 2 belong to the Mandelbrot set, then the quartic polynomial obtained in Theorem A satisfies the assumptions of Theorem 1.4.

2.
Construction of the desired quartic polynomials. In this section, we construct the desired quartic polynomial f and prove Theorem A. The construction uses Lemma 2.1 and Theorem 2.2 on quasiregular mappings. In Lemma 2.1, "log" denotes the principal branch of the logarithm.
Lemma 2.1 ([4, 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 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 2.2 (Quasiconformal surgery [4, 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; Notations. Let R > r > 0, A ∈ C \ {0} and c ∈ C. We use the notations below.
The quadratic polynomial q c is affine conjugate to p c . The quartic polynomial h 1 and h 2 are affine conjugate to h. It is easy to check that critical points of h are 0 and ±R 2 . Hence, critical points of h 1 are 0, R 2 and 2R 2 . Similarly, critical points of h 2 are 0, −R 2 and −2R 2 . For the sake of convenience, we deal with the quartic polynomials h 1 and h 2 simultaneously as Lemma 2.3. Let k = 2, R 1 = r, R 2 = R, ϕ 1 = q c and ϕ 2 = h . If A = 4R 4 , r = R/2 and R is large enough, then the inequality ( * ) of Lemma 2.1 holds for δ 0 = (log 2)/3.
Lemma 2.7. Let r = R/2. If R is large enough, then the orbit of any point in Γ 1 ∪ Γ 2 tends to infinity or h n re iy ∓ R 2 → ∞.

Proof.
Since holds if R is large enough. By Lemma 2.6, we obtain the result.
If R is large enough, the critical orbit h n (0) = h n+1 (±R 2 ) tends to infinity. Therefore, G(0) and G(±R 2 ) are positive, where G = G h is the Green's function associated with the filled-in Julia set K(h). Since the preimages of the critical value R 8 = h(0) are 0 and ± √ 2R 2 , then the equipotential curve Φ = G −1 (G(0)) is a figure-eight through 0 and ± √ 2R 2 . It has symmetry with respect to the origin because h is even. Moreover, since critical points ±R 2 are the preimages of the critical point 0, then the equipotential curve G −1 (G(R 2 )) = G −1 (G(−R 2 )) has two components Θ 1 and Θ 2 , which are the congruent quatrefoils centered at −R 2 and R 2 respectively. The quatrefoils Θ 1 and Θ 2 are surrounded by the figure-eight Φ. Since the critical orbit h n (0) = h n+1 (±R 2 ) tends to infinity for sufficiently large R, the Julia set J(h) is totally disconnected and it is surrounded by Θ 1 ∪ Θ 2 . By Lemma 2.7, it is also surrounded by Γ 1 ∪ Γ 2 .
The strategy of the proof of Theorem A.
(1) Take R sufficiently large and cut off the Julia set J(h) by the two circles Γ 1 and Γ 2 . (2) Paste two copies of the quadratic Julia sets J(p c1 ) and J(p c2 ) in the interior of the circles Γ 1 and Γ 2 respectively. (3) In order to construct a quasiregular map g, interpolate h and two quadratic polynomials which are hybrid equivalent to p c1 and p c2 respectively. (4) To obtain the desired quartic polynomial f , employ the quasiconformal surgery for g. for j = 1 and 2. We take R sufficiently large such that Γ surrounds the filled-in Julia sets K(q c1 ) and K(q c2 ). We define a quasiregular map g : C → C as follows : We check the assumptions of Theorem 2.2. Let If R is large enough, then For z = re iy ∓ R 2 ∈ Γ 1 ∪ Γ 2 , and the orbit of any point in E 1 ∪ E 2 under g tends to infinity, which implies that the assumption (1) of Theorem 2.2 holds. The other assumptions (2), (3) and (4) of Theorem 2.2 obviously hold. Therefore, there exists a 4-quasiconformal map ϕ : C → C such that f = ϕ • g • ϕ −1 is an entire function of degree 4. Hence, f is a quartic polynomial. We normalize ϕ as ϕ(0) = 0 and ϕ(1) = 1. Then the finite critical points of f are 0 and ϕ(±R 2 ).
Let U 1 and U 2 be the bounded components of C\G −1 (G(0)) such that ϕ(−R 2 ) ∈ U 1 and ϕ(R 2 ) ∈ U 2 , where G = G f is the Green's function associated with the filledin Julia set K(f ). Let V be the bounded component of C \ G −1 (G(f (0))). Then (f | U1 , U 1 , V ) and (f | U2 , U 2 , V ) become quadratic-like maps. By the straightening theorem, the quadratic-like map (f | Uj , U j , V ) is hybrid equivalent to the quadratic polynomial q cj , which is affine conjugate to the quadratic polynomial p cj for j = 1 and 2.