Weiyuan Qiu

Pascale Roesch

ESCAPE COMPONENTS OF MCMULLEN Yueyang Wang

Pascale Roesch

Pacale Roesch

ESCAPE COMPONENTS OF MCMULLEN MAPS

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

f λ (z) = z n + λ z n with λ ∈ C * and n ≥ 3. This family has been widely studied since it exhibits a very rich dynamical behavior. In this article, we focus our attention on the escape locus in the parameter plane of this family which consists of those maps such that all critical orbits are attracted by the supper-attracting fixed point ∞. Every connected component of the escape locus is a hyperbolic component, called an escape component. It was proved in [START_REF] Qiu | Hyperbolic components of McMullen maps[END_REF] that every escape component is bounded by a Jordan curves. It can also be shown that each hyperbolic component outside the escape locus belongs to a homeomorphic copy of the Mandelbrot set.

In this article, we explore the geometric regularity of the boundaries of escape components. There is a unique unbounded escape component which is called the Cantor locus. Like in the case of polynomials, a map in the Cantor locus has all of its critical values in the Fatou component of ∞. Hence its Julia set is a Cantor set. In [START_REF] Qiu | Hyperbolic components of McMullen maps[END_REF], the authors proved that cusps are dense on the boundary of the Cantor locus. In the bounded escape components, the McMullen maps behavior more like rational maps. A map in a bounded escape component has its critical value in a strict pre-image of the Fatou component of ∞. There is a unique escape component centered at the origin which is called the McMullen domain such that a map in it have its Julia sets homeomorphic to a Cantor set × circle. All the other escape components are called Sierpiński holes since maps in them have Julia sets as Sierpiński carpets. The main purpose of this article is to characterize the geometric property of all bounded escape components. We obtain the following result.

Theorem 1.1. The boundary of each bounded escape component is a quasi-circle with Hausdorff dimension strictly between 1 and 2.

We show that these geometric properties can be deduced from the following more precise topological characterization about the closures of escape components. The key step is to show that the boundary of each bounded escape component does not intersect with the boundary of the Cantor locus. To prove this, we first construct a local para-puzzle system in the parameter space which is the counterpart of dynamical puzzle system introduced in [START_REF] Qiu | Dynamics of McMullen maps[END_REF][START_REF] Qiu | Hyperbolic components of McMullen maps[END_REF]. With the aid of this para-puzzle system, we can characterize the dynamical behavior of the hypothetical intersection points. We exclude all possibilities except the intersections at parabolic parameters. Then we use the parabolic implosion theory introduced in [Shi00, Lei00] to exclude the parabolic case. Indeed, on one hand the intersection point could be accessed by a parameter ray which consists of maps with critical values in a strict pre-image of Fatou component of ∞. On the other hand, according to the parabolic implosion theory, the Fatou coordinate remains stable under a perturbation within a particular sector. After showing that the parameter ray is contained in the sector, we conclude that the maps on this parameter ray should have critical points in the Fatou component of ∞ (not in a strict pre-image of Fatou component of ∞). This is a contradiction. Finally we show that the closures of bounded escape components are pairwise disjoint. One hidden difficulty is to exclude the intersection between two bounded escape components with potentially the same dynamical property. We solve this by establishing a rigidity result of bounded escape components.

To prove Theorem 1.1, we first observe the boundary of the Fatou component of ∞ is a quasi-circle with Hausdorff dimension strictly between 1 and 2 for the map does not belong to the closure of the Cantor locus. Then Theorem 1.2 allows us to transfer this property to the boundaries of bounded escape components in the parameter space via a holomorphic motion. (2.1) f λ (z) = z n + λ z m with λ ∈ C * = C \ {0} and n ≥ 2, m ≥ 1. They showed that this family exhibits a very rich dynamical behavior while it has such a simple form. In [START_REF] Robert L Devaney | The escape trichotomy for singularly perturbed rational maps[END_REF], the authors showed that if the free critical orbits escape to ∞, then its Julia set is either a Cantor set, a Cantor circle or a Sierpiński carpet. Later on, R. Devaney and his group published a variety of papers on this McMullen family see [Dev07, DM07, DP09, Dev04, DL05, Dev05, DLU05, Dev06, Dev08, Dev13]. Among these articles, lots of them devoted to the particular case of n = m in equation (2.1). i.e.

(2.2)

f λ (z) = z n + λ z n with λ ∈ C * = C \ {0}
and n ≥ 3. This is also the family we study in this article.

In [START_REF] Qiu | Dynamics of McMullen maps[END_REF], the authors used the cut rays introduced in [START_REF] Robert | Cantor necklaces and structurally unstable Sierpinski curve Julia sets for rational maps[END_REF] to construct a puzzle system and showed that the boundary of the immediate basin of ∞ is either a Cantor set or a Jordan curve. The whole Julia set is locally connected except for some special cases. Later, they further developed the rigidity theory via this puzzle system to study the boundaries of hyperbolic components in the parameter space, see [START_REF] Qiu | Hyperbolic components of McMullen maps[END_REF]. They proved that all hyperbolic components of the McMullen family in equation (2.2) are bounded by Jordan curves.

The McMullen maps admit a symmetric conjugacy

(2.3) e 2πi n-1 f λ (z) = (-1) n f e 2πi n-1 λ e 2πi n-1 z
for all λ ∈ C * . Hence it suffices to study maps whose parameters belong to the sector (2.4)

F 0 := λ ∈ C * : 0 ≤ arg λ ≤ 2π n -1 .
We denote the interior of F 0 as F . The critical set of

f λ is {0, ∞} ∪ C λ , where C λ = {c ∈ C : c 2n = λ}.
Here, ∞ is a supper-attracting fixed point of f λ which has only two pre-images 0 and ∞. There are only two critical values

v ± λ = ±2 √ λ other than ∞ (here v ± λ is well-defined for λ ∈ F 0 , v + λ is defined to be the one belongs to {z ∈ C : 0 ≤ arg z < π}). Let (2.5) Z k (λ) := {∞}, k = 0, z ∈ C : f k-2 λ (z) = 0 , k ≥ 2. For k ≥ 2, Z k (λ) is the set of all (k -1)-th iterated pre-images of ∞ under f λ with ∞ itself excluded. Lemma 2.1 (Böttcher coordinate). For each z k (λ) ∈ Z k (λ), the Böttcher coordinate φ z k (λ) near z k (λ) is defined in the following. (1) For k = 0, z 0 (λ) = ∞, φ ∞(λ) is defined to be (2.6) φ ∞(λ) (z) = lim k→∞ n k f k λ (z) with φ ∞(λ) (∞) = 1. It satisfies the equation φ ∞(λ) (f λ (z)) = φ ∞(λ) (z) n and φ ∞(λ) (e πi/n z) = e πi/n φ ∞(λ) (z). (2) For k = 2, z 2 (λ) = 0, φ 0(λ) is defined to be φ 0(λ) := n φ ∞(λ) • f λ which satisfies that φ 0(λ) (0) = n √ λ. (3) For k ≥ 3 and z k (λ) ∈ Z k (λ) \ l≥0 f -l λ (C λ ), φ z k (λ) is defined to be φ z k (λ) := φ 0(λ) • f k-2 λ .
Remark 2.1. Let D(a, r) denote the disk with center at a and radius r, and let D r = D(0, r). By Section 9 in [START_REF] Milnor | Dynamics in One Complex Variable[END_REF], φ -1 z k (λ) can be extended on a maximal disk

C\D sz k (λ) with s z k (λ) = 1 or with s z k (λ) > 1 such that ∂φ -1 z k (λ) (D sz k (λ) ) ∩ l≥0 f -l λ (C λ ) = ∅. Each Böttcher coordinate φ z k (λ) introduces a system of dynamical rays and equipo- tential curves near z k (λ). The dynamical ray R t z k (λ) with angle t ∈ R/Z is defined to be R t z k (λ) := φ -1 z k (λ) ((s z k (λ), ∞]e 2πit ). The equipotential curve E s z k (λ) with s ≥ s z k is defined to be E s z k (λ) := φ -1 z k (λ) se 2πiR/Z
. Consider the subset of the parameter plane, called the escape locus, which consists of parameters of maps whose all critical orbits escape to infinity. That is,

H := λ ∈ C * : lim k→∞ f k λ (C λ ) = ∞ .
The escape locus H is an open subset of the parameter plane, maps inside H are called escape maps. Connected components of H are called escape components. It is worth mentioning that all escape components are hyperbolic components. According to the discussion in [START_REF] Qiu | Hyperbolic components of McMullen maps[END_REF], hyperbolic components not in H has "renormaliziable" type. It could be proved that all hyperbolic components of this type belong to small copies of the Mandelbrot set, see Figures 1,2. Let B λ denotes the Fatou component containing ∞ and T λ denotes the Fatou component containing 0, which is the unique component of f -1 λ (B λ ) different from B λ itself. We can further distinguish those escape components by how many iterations are needed for the critical set C λ mapped into B λ . For λ ∈ H , we define its order N (λ) to be:

N (λ) = min k ∈ N : f k λ (C λ ) ⊂ B λ .
It is obvious that the order N (λ) take constant value on each component of H . Hence we can decompose the escape locus such that H = k≥0 H k , where H k = {λ ∈ H : N (λ) = k} consists of all escape maps with order k.

Remark 2.2. It is known that λ ∈ H 0 if and only if v ± λ ∈ B λ , H 1 = ∅, and λ ∈ H k for k ≥ 2 if and only if f k-1 λ (v ± λ ) ∈ T λ .
According to [START_REF] Robert L Devaney | The escape trichotomy for singularly perturbed rational maps[END_REF] and [START_REF] Roesch | On capture zones for the family f c (z) = z 2 + c/z 2 . Dynamics on the Riemann Sphere[END_REF][START_REF] Steinmetz | On the dynamics of the McMullen family[END_REF], we have the following escape trichotomy and parameterization for escape components.

Theorem 2.2 (Escape trichotomy and parameterization). We have the following trichotomy and the parameterization.

(1) The Cantor locus H 0 is the unique unbounded hyperbolic component. The parameterization map Φ H 0 :

H 0 → C \ D defined by Φ H 0 (λ) := φ ∞(λ) (v + λ ) 2 is a holomorphic homeomorphism. For λ ∈ H 0 , its Julia set J(f λ ) is a Cantor set, see Figure 3. (2) The McMullen domain H 2 is the unique hyperbolic component containing 0. The parameterization map Φ H 2 : H 2 → C \ D defined by Φ H 2 (λ) n-2 := φ ∞(λ) • f λ (v + λ ) 2 is a holomorphic homeomorphism with lim λ→0 λΦ H 2 (λ) = 2 2n n-2 . For λ ∈ H 2 , its Julia set J(f λ ) is a Cantor circle, see Figure 4.
(3) Sierpiński holes are connected components of H k with order k ≥ 3. For each Sierpiński hole U , the parameterization map

Φ U : U → C \ D defined by Φ U (λ) := φ 0(λ) • f k-2 λ (v + λ ) is a holomorphic homeomorphism. For λ ∈ U , its Julia set J(f λ ) is a Sierpiński carpet, see Figure 5.
Remark 2.3. Hyperbolic components not in H are called non-escape components. All non-escape hyperbolic components are contained in small copies of the Mandelbrot set. Map belongs to a non-escape hyperbolic component has its critical orbit k≥0 f k λ (C λ ) attracted by attracting periodic orbits different from ∞, see Figure 6. It is easy to verify that for λ ∈ H 0 , (2.7)

Φ H 0 e 2πi n-1 λ = e 2πi n-1 Φ H 0 (λ) and (2.8) Φ H 0 λ = Φ H 0 (λ).
Let U ⊂ H be an escape component. The parameter ray R t U in U with angle t ∈ R/Z is defined to be R t

U := Φ -1 U ((1, ∞)e 2πit ). The equipotential curve E s U in U with level s is defined to be E s U := Φ -1 U se 2πiR/Z .
2.2. Dynamical puzzles. Let Λ ⊂ C be a hyperbolic region and X be a subset of C. Let π 1 : Λ × X → Λ and π 2 : Λ × X → X be two projection maps defined by π 1 (λ, z) = λ and π 2 (λ, z) = z respectively. A holomorphic motion of X, parameterized by Λ, with the base point at

λ 0 ∈ Λ is a map h : Λ × X → Λ × C such that (1) for each x ∈ X, the map λ → π 1 • h(λ, x) is the identity map. (2) for each x ∈ X, the map λ → π 2 • h(λ, x) is holomorphic, (3) for each λ ∈ Λ, the map x → π 2 • h(λ, x) is injective, (4) the map x → π 2 • h(λ 0 , x) is the identity map. Let us denote h λ : X → C for the map x → π 2 • h(λ, x).
Theorem 2.3 (Slodkowski [START_REF] Slodkowski | Extensions of holomorphic motions[END_REF][START_REF] Douady | Prolongement de mouvements holomorphes[END_REF] ). Suppose h : Λ × X → Λ × C is a holomorphic motion, then h can be extended to a holomorphic motion h : Λ × C → Λ × C. For the extended holomorphic motion h, for each λ ∈ Λ, the map h λ : C → C is a quasi-conformal homeomorphism. Furthermore, the Beltrami coefficient µ λ = ∂h λ /∂h λ of h λ satisfies (2.9)

µ λ ∞ = ess sup x∈X |µ λ (x)| ≤ e ρ(λ,λ 0 ) -1 e ρ(λ,λ 0 ) + 1 ,
where ρ(λ, λ 0 ) is the hyperbolic distance between λ and λ 0 in Λ.

Let τ : R/Z → R/Z defined by τ (θ) = nθ mod Z.

Let Θ k = k 2n , k+1 2n for 0 ≤ k ≤ n and Θ -k = k 2n + 1 2 , k+1 2n + 1 2 for 1 ≤ k ≤ n -1.
Obviously, (0, 1] = -n<j≤n Θ j . Let Θ be the set of all angles θ ∈ (0, 1] whose orbits remain in

n-1 k=1 (Θ k ∪ Θ -k ) under iterations of τ . Let Θ per = p≥1 {θ ∈ Θ : τ p θ = θ} \ 1, 1 2 .
Then Θ is a Cantor set and Θ per is a dense subset of Θ. Following [START_REF] Robert | Cantor necklaces and structurally unstable Sierpinski curve Julia sets for rational maps[END_REF][START_REF] Qiu | Hyperbolic components of McMullen maps[END_REF], for each θ ∈ Θ, there is a cut ray Ω θ λ which cut the Julia set into two parts. Instead of giving the precise definition of cut rays, we conclude the core properties of cut rays in the following Theorem 2.4. It is the combination of Theorem 3.2, Lemma 3.3, Theorem 3.4 in [START_REF] Qiu | Hyperbolic components of McMullen maps[END_REF].

Theorem 2.4 (Properties of cut rays). For any θ ∈ Θ per and λ ∈ F 0 , the cut ray Ω θ λ with angle θ is a Jordan curve containing 0 and ∞ and symmetric with respect to 0 satisfies the following properties:

(1)

f p λ (Ω θ λ ) ⊂ Ω θ λ , where p is the period of θ, (2) Ω θ λ ∩ J(f λ ) is a Cantor set, and Ω θ λ ∩ B λ = R θ ∞(λ) ∪ R θ+1/2 ∞(λ) , (3) fix any λ 0 ∈ F , there is a holomorphic motion h : F × Ω θ λ 0 → F × C based at λ 0 such that h λ (Ω θ λ 0 ) = Ω θ λ , (4
) fix any λ 0 ∈ R + , there exists a neighborhood W θ containing R + and a holomorphic motion h :

W θ × Ω θ λ 0 → W θ × C based at λ 0 such that h λ (Ω θ λ 0 ) = Ω θ λ .
If for some N ≥ 1, Ω θ λ \ {0, ∞} ∩ 1≤k≤N f k λ (C λ ) = ∅, then for any α ∈ 0≤k≤N τ -k (θ), there is a unique Jordan curve Ω α λ containing 0 and ∞ such that

f λ (Ω α λ ) = Ω τ (α) λ and R α ∞(λ) ∪ R α+1/2 ∞(λ) = Ω α λ ∩ B λ .
Under this circumstance, the Jordan curve Ω α λ is also called a cut ray. Lemma 2.5 ([QRWY15] Lemma 3.7 ). Let λ ∈ F 0 and R t 0 ∞(λ) and R t 1 ∞(λ) be two dynamical rays with distinct angles t 0 and t 1 . If

(2.10) Ω θ λ \ {0, ∞} ∩ k≥1 f k λ (C λ ) = ∅ for θ ∈ Θ per , then there is a cut ray Ω α λ with angle α ∈ k≥0 τ -k (θ) separating R t 0 ∞(λ)
and R t 1 ∞(λ) . In [START_REF] Qiu | Dynamics of McMullen maps[END_REF], the authors used cut rays to construct a puzzle system for the McMullen family and studied the locally connectivity of Julia sets. Puzzles is a regular tool in studying holomorphic dynamics, see [Hub92, BH92, Roe07, Sch04, KL09, QY09, Roe08, LvS98, Pet96, KSvS07].

For any L > 1, denote

X L λ := {z ∈ B λ : |φ ∞(λ) (z)| > L}. Given a parameter λ ∈ F 0 , we can find Θ λ = {θ 1 , θ 2 , • • • , θ N } ⊂ Θ per and L > 1 such that each cut ray Ω θ λ with θ ∈ Θ λ is well-defined and C λ ∩ X L λ = ∅.
In the dynamical plane of f λ , the graph of depth 0 associated with Θ λ is defined to be

I 0 λ (Θ λ ) = ∂X L λ ∪ C \ X L λ ∩ m≥0 Ω τ m (θ 1 ) λ ∪ Ω τ m (θ 2 ) λ ∪ • • • ∪ Ω τ m (θ N ) λ .
The graph of depth k is defined to be

I k λ = f -k λ (I 0 λ (Θ λ )). A puzzle piece P k λ of depth k is a connected component of f -k λ ((C \ X L λ ) \ I 0 λ ). The puzzle piece of depth k containing z ∈ J(f λ ) is denoted by P k λ (z).
In [START_REF] Qiu | Dynamics of McMullen maps[END_REF][START_REF] Qiu | Hyperbolic components of McMullen maps[END_REF], the authors used the puzzle system (f λ , I k λ , P k λ ) to study the dynamics of McMullen maps and obtained the following results (Theorems 1.1, 1.2, 1.4 in [START_REF] Qiu | Dynamics of McMullen maps[END_REF] and Theorem 1.1 in [START_REF] Qiu | Hyperbolic components of McMullen maps[END_REF]).

Theorem 2.6.

For the McMullen family f λ (z) = z n + λ z n with n ≥ 3 and λ ∈ C * . (1) If λ / ∈ H 0 , then ∂B λ is a Jordan curve. Furthermore, if ∂B λ contains neither a parabolic point nor a recurrent critical point, then ∂B λ is a quasi-circle. (2) Suppose U ⊂ H k is an escape component with k ≥ 2 and λ ∈ ∂U \ ∂H 0 , then for each z ∈ J(f λ ), k≥0 P k λ (z) = {z} and J(f λ ) is locally-connected. (3) Suppose U ⊂ H is an escape component, then ∂U is a Jordan curve.
By part 1 of Theorem 2.6, for λ / ∈ H 0 , the Böttcher coordinate φ ∞(λ) can be extended to a homeomorphism φ ∞(λ) : B λ → C \ D. By part 3 of Theorem 2.6, for each escape component U ⊂ H , the parameterization map Φ U defined in Theorem 2.2 can be extended to a homeomorphism on its closure.

Local para-puzzles

In this section, we construct a local-puzzle system at λ 0 ∈ F \ H to study the bifurcation of the puzzle systems (f λ , I k λ , P k λ ) when the parameter λ various near λ 0 . This is a local version of the para-puzzle constructed for polynomials, see [Roe00, Lyu00, Hub92, Fau93, Roe07, ALS10].

Proposition 3.1 (Existence of para-puzzles). For each λ 0 ∈ F 0 \ H , there exists a sequence of simply connected open neighborhoods {P k λ 0 } k≥0 of λ 0 , which are called para-puzzle pieces, such that the following holds.

(1) P 0

λ 0 ⊃ P 1 λ 0 ⊃ • • • ⊃ P k λ 0 ⊃ • • • ⊃ {λ 0 }. (2) For each k ≥ 0, there exists a holomorphic motion H k : P k λ 0 ×I k λ 0 → P k λ 0 ×C such that for each λ ∈ P k λ 0 , H k λ, I k λ 0 = λ, I k λ .
Proof. Proposition 3.1 will be proved by induction. By part 3 and part 4 of Theorem 2.4, there exists a simply-connected region W containing λ 0 such that h : W ×

θ∈Θ λ 0 Ω θ λ 0 → W × C is a holomorphic motion. It is not hard to find a simply- connected region P 0 λ 0 ⊂ W containing λ 0 such that the map H 0 : P 0 λ 0 × I 0 λ 0 → P 0 λ 0 × C defined by (3.1) H 0 (λ, z) := h(λ, z), (λ, z) ∈ P 0 λ 0 × θ∈Θ λ 0 Ω θ λ 0 , λ, φ -1 ∞(λ) • φ ∞(λ 0 ) (z) , (λ, z) ∈ P 0 λ 0 × ∂X L λ 0
is a well-defined holomorphic motion which satisfies H 0 λ, I 0 λ 0 = (λ, I 0 λ ). Assume the para-puzzle pieces P m λ 0 and the holomorphic motion

H m : P m λ 0 ×I m λ 0 → P m λ 0 × C are already constructed for 0 ≤ m ≤ k -1.
The local parameter graph of depth k -1 in the para-puzzle piece P k-1 λ 0 is defined to be

I k-1 := λ ∈ P k-1 λ 0 : v + λ ∈ I k-1 λ .
By the construction of puzzle system (f

λ 0 , I k λ 0 , P k λ 0 ), v + λ 0 / ∈ I k-1 λ 0 . Hence λ 0 / ∈ I k-1 . Let λ j → λ ∈ P k-1 λ 0 as j → ∞ with {λ j : j ≥ 1} ⊂ I k-1 . Since H k-1 is a holomorphic motion, v + λ j ∈ I k-1 λ j for each j ≥ 1 implies that v + λ ∈ I k-1 λ . Hence I k-1 is relatively closed in P k-1 λ 0 . Define P k λ 0 to be a simply connected open subset of P k-1 λ 0 \ I k-1 containing λ 0 (P k λ 0 = P k-1 λ 0 if I k-1 = ∅). Since v + λ / ∈ I k-1 λ for λ ∈ P k λ 0
, there exists a holomorphic motion H k such that the following diagram commutes (3.2)

P k λ 0 × I k λ 0 H k ---→ P k λ 0 × C   i.×f λ 0   i.×f λ P k-1 λ 0 × I k-1 λ 0 H k-1 ---→ P k-1 λ 0 × C, where i. : P k λ 0 → P k-1 λ 0 denotes the inclusion map. By diagram (3.2), H k λ, I k λ 0 = λ, I k λ .
Hence the Proposition 3.1 holds for m = k. Corollary 3.2. There exists a holomorphic motion h k :

P k λ 0 × I k-1 λ 0 ∪ v + λ 0 → P k λ 0 × C such that h k | P k λ 0 ×I k-1 λ 0 = H k-1 and h k λ, v + λ 0 = λ, v + λ .
Proof. This follows immediately from that v + λ / ∈ I k-1 λ for λ ∈ P k λ 0 .

Centers of H

In this section, we discuss the relation between the centers of hyperbolic components of H and the iterated pre-images of ∞ in the dynamical plane.

Suppose that

U k is a connected component of H k , k ≥ 2. Recall the param- eterization map Φ U k defined in Theorem 2.2. The center of U k is defined to be λ U k := Φ -1 U k (∞) (for the McMullen domain H 2 , define λ H 2 := 0) which satisfies the equation f k-2 λ (v + λ ) = 0. Let Λ k := λ ∈ C * : f k-2 λ (v + λ ) = 0 . Then Λ k is a finite set since f k-2 λ (v + λ ) = 0 is an algebraic equation. It follows that H k has finitely many connected components since the number of connected components of H k is equal to #Λ k . Lemma 4.1. Let V ⊂ C * \ 2≤j≤k-1 Λ j be a simply-connected region, then there exist (2n) k-2 distinct holomorphic functions z i k (λ) defined on V such that f k-2 λ (z i k (λ)) = 0, i = 1, 2, . . . , (2n) k-2 .
Proof. Consider the algebraic equation

(4.1) f k-2 λ (z) = 0. For k = 2, Lemma 4.1 is trivial. For k ≥ 3, since V ⊂ C * \ 2≤j≤k-1 Λ j , for each λ ∈ V , the solution z of equation (4.1) can not be a critical point of f k-1 λ . For otherwise, by f k-2 λ (z) = 0≤i≤k-3 (f λ ) f i λ (z) , there is an 0 ≤ i ≤ k -3 such that f i λ (z) ∈ C λ which implies f i+1 λ (z) = v ± λ . Hence f k-3-i λ (v + λ ) = 0, which implies that λ ∈ Λ k-i-1 which is a contradiction. It follows that equation (4.1) has (2n) k-2 distinct roots z i for 1 ≤ i ≤ (2n) k-2 . By implicit function theorem, there exist (2n) k-2 distinct holomorphic functions z i k (λ) defined near λ 0 such that f k-2 λ (z i k (λ))=0 and z i k (λ 0 ) = z i for 1 ≤ i ≤ (2n) k-2 . Since V is simply-connected, the (2n) k-2 holomorphic functions z i k (λ) can be extended to the whole region V . Define z 0 : C * → C to be the constant map z 0 (λ) = ∞. z 0 and each z i k for 1 ≤ i ≤ (2n) k-2 defined in Lemma 4.1 are called root functions.
Recall the definition of Thurston's combinatorial equivalence and Thurston's rigidity theorem, see [START_REF] John | Teichmüller theory and applications to geometry, topology, and dynamics[END_REF], [START_REF] Douady | A proof of Thurston's topological characterization of rational functions[END_REF].

Definition 4.2 (Thurston's combinatorial equivalence). Let f, g : S 2 → S 2 are both post-critically finite branched covering with post-critical sets P (f ) and P (g). If there exist two orientation preserving homeomorphism φ, ψ from S 2 to itself such that the following diagram commutes.

(4.2) (S 2 , P (f )) ψ ---→ (S 2 , P (g))   f   g (S 2 , P (f )) φ ---→ (S 2 , P (g))
Furthermore, φ and ψ satisfies that φ| P (f ) = ψ| P (f ) and φ and ψ are isotopic to each other relatively to the post-critical set P (f ). Then we say the maps f, g are Thurston's combinatorial equivalent.

Remark 4.1. If φ and ψ are both quasi-conformal homeomorphisms, then the condition of isotopy in definition 4.2 can be replaced by homotopy, see [START_REF] John | Teichmüller theory and applications to geometry, topology, and dynamics[END_REF].

Theorem 4.3 (Thurston's Rigidity). If two post-critically finite rational maps f, g are Thurston combinatorial equivalent, then f, g are conformally conjugated.

Proposition 4.4. Let k ≥ 3 and V ⊂ C * \ j≤k-1 Λ j be a simply-connected region. For λ 1 , λ 2 ∈ V ∩ F 0 , if there exists a root function z i k such that z i k (λ 1 ) = v + λ 1 and z i k (λ 2 ) = v + λ 2 , then λ 1 = λ 2 . Proof. Let W := λ ∈ C * : f l λ v + λ , 0 ≤ l ≤ k -3, are k -2 distinct points . It is obvious that C * \ W is a finite set, we may assume that V ⊂ W . Define Φ : V × P (f λ 1 ) → V × C by (4.3) Φ λ, f l λ 1 v ± λ 1 := (λ, f l λ (v ± λ )), 0 ≤ l ≤ k -3, (λ, id.), l ≥ k -2.
It is easy to check that f l λ v ± λ : 0 ≤ l ≤ k -3 ∪ {0, ∞} contains exactly 2k -2 (or k + 1) distinct points for λ ∈ W \ j≤k-1 Λ j if n is odd (or even). Each of one is holomorphically depending on the parameter λ. This implies that Φ is a holomorphic motion based at λ 1 . By Slodkowski's Theorem 2.3, Φ can be extended to a holomorphic motion Φ :

V × C → V × C. Let F (λ, z) := (λ, f λ (z)) for (λ, z) ∈ V × C.
Claim 1. There exists a lifted mapping Ψ : V × C → V × C of Φ which is also a holomorphic motion such that the following diagram commutes.

(4.4)

V × C Ψ ---→ V × C   id×f λ 1   F V × C Φ ---→ V × C.
Furthermore, Ψ satisfies the following property on the post-critical set

P (f λ 1 ) of f λ 1 (4.5) Ψ λ, f l λ 1 v ± λ 1 =      λ, f l λ v ± λ 0 ≤ l ≤ k -4, λ, f l λ (±z i k (λ)) l = k -3, (λ, id.) l ≥ k -2. Proof. Since the map Φ(λ, v ± λ 1 ) = (λ, v ± λ ) and Φ(λ, ∞) = (λ, ∞)
, there exists a unique lifting map Ψ : V × C → V × C such that it is a holomorphic motion and satisfies the diagram (4.4).

For

l = k -3, denote h ± (λ) = Ψ λ • f k-3 λ 1 v ± λ 1 . By (4.3) and diagram (4.4), f λ (h ± (λ)) = f k-2 λ 1 (v ± λ 1 ) = 0. Since z i k (λ 1 ) = v + λ 1 , then h ± (λ 1 ) = f k-3 λ 1 v ± λ 1 = f k-3 λ 1 (±z i k (λ 1 ))
. By Lemma 4.1, there are 2n distinct pre-images of 0 which are holomorphic with respect to λ ∈ V . Hence h

± (λ) = f k-3 λ (±z i k (λ)) for λ near λ 1 . Since V is connected, then h ± (λ) = f k-3 λ (±z i k (λ)
) for all λ ∈ V . The proofs for l ≤ k -4 and l ≥ k -2 are very similar and will be omitted here.

The proof of Proposition 4.4 continues. By Claim 1 and the condition

z i k (λ 2 ) = v + λ 2 , we have Ψ λ 2 • f k-3 λ 1 (v ± λ 1 ) = f k-3 λ 2 (±z i k (λ 2 )) = f k-3 λ 2 v ± λ 2 . Hence Φ λ 2 • f l λ 1 (v ± λ 1 ) = Ψ λ 2 • f l λ 1 (v ± λ 1 ) = f l λ 2 v ± λ 2 for all l ≥ 0. By diagram (4.4), we have (4.6) C Ψ λ 2 ---→ C   f λ 1   f λ 2 C Φ λ 2 ---→ C. Let H : [0, 1] × C → C defined by H t = (1 -t) Φ λ 2 + tΨ λ 2 .
It is easy to check that H 1 = Ψ λ 2 , H 0 = Φ λ 2 and H t | P (f λ 1 ) = Ψ λ 2 . Hence Ψ λ 2 and Φ λ 2 are homotopic relative to P (f λ 1 ). Since Φ λ 2 and Ψ λ 2 are both quasi-conformal homeomorphism, by Remark 4.1 and Thurston's rigidity Theorem 4.3, f λ 1 and f λ 2 are conformally conjugated to each other. Since λ 1 , λ 2 ∈ F 0 , then it is easy to check that λ 1 = λ 2 .

Corollary 4.5. Let U ⊂ H k be an escape component and λ = Φ -1 U (re 2πit )∩F 0 . Let V ⊂ C * \ j≤k-1 Λ j be a simply-connected region containing U , then there exists a unique root function t) where

z i k (λ) defined on V such that v + λ = φ -1 z k (λ) ρ k (r)e 2πiθ k (
(4.7) θ k (t) :=      t 2 , k = 0, 2nt n-2 , k = 2, t, k ≥ 3 and (4.8) ρ k (r) :=    √ r, k = 0, n-2 √ r 2n , k = 2, r, k ≥ 3.
Proof. First, let us show the existence and uniqueness of the root function. Since

f k-2 λ v + λ U
= 0, by Lemma 4.1 and Proposition 4.4, there exists a unique root function

z i k (λ) defined on V such that z i k (λ U ) = v + λ U (the uniqueness is trivial for k ≤ 2 and insured by Proposition 4.4 for k ≥ 3). It follows easily that if λ ∈ U , v + λ is contained in the Fatou component containing z i k (λ)
. By Remark 2.1, the Böttcher coordinates φ z i k (λ) defined in Lemma 2.1 can be extended to v + λ . The rest of the proof follows easily from the asymptotic behavior of the parameterization maps defined in Theorem 2.2 and dynamical Böttcher coordinates defined in Lemma 2.1.

Dynamics of maps on ∂H

Let U ⊂ H be an escape component. In this section, we describe the dynamics of f λ for λ = Φ -1

U (e 2πit ) ∈ ∂U ∩F 0 where t ∈ R/Z via the puzzle system (f λ , I k λ , P k λ ). Lemma 5.1. If λ ∈ F 0 \ H 0 , then for any sequence of puzzle pieces {P k λ } k≥0 , k≥0 P k λ ∩ ∂B λ is either empty or a singleton. Proof. Suppose k≥0 P k λ ∩ ∂B λ = ∅. Then {P k λ } is a nested sequence of puzzle pieces. By part 1 in Theorem 2.6, every points in

k≥0 P k λ ∩ ∂B λ is a landing point of a dynamical ray. Suppose that φ -1 ∞(λ) (e 2πit 0 ) ∈ k≥0 P k λ ∩ ∂B λ . It suffices to show for t = t 0 , φ -1 B λ (e 2πit ) / ∈ k≥0 P k λ ∩ ∂B λ .
Since each θ ∈ Θ λ satisfies (2.10) in Lemma 2.5, by the construction of graphs, there is an angle α ∈ k≥0 τ -k (θ) such that the cut ray Ω α λ separates φ -1 ∞(λ) (e 2πit ) and φ -1 ∞(λ) (e 2πit 0 ). Note that there must be a k such that α ∈ τ -k (θ) which implies that Ω α λ ⊂ I k λ . We get that φ -1 ∞(λ) (e 2πit ) and φ -1 ∞(λ) (e 2πit 0 ) are separated by the graph

I k λ of depth k. This follows that φ -1 ∞(λ) (e 2πit ) / ∈ P k λ φ -1 ∞(λ) (e 2πit 0 ) = P k λ , hence φ -1 ∞(λ) (e 2πit ) / ∈ k≥0 P k λ ∩ ∂B λ . Let us denote K ± λ := k≥0 P k λ (v ± λ ). Proposition 5.2. Let U ⊂ H k be an escape component with k ≥ 0. If λ 0 = Φ -1 U (e 2πit ) ∈ F 0 , then there exists z i k (λ 0 ) ∈ Z k (λ 0 ) such that w t k (λ 0 ) := lim s→1 φ -1 z i k (λ 0 ) se 2πiθ k (t) ∈ K + λ 0 ,
where θ k (t) is defined in (4.7).

Proof.

For k = 0, z i k (λ 0 ) = ∞. For k ≥ 2, there is a simply-connected region U * ⊃ U such that U * ∩ l≤k-1 Λ l = ∅. Let z i k (λ)
denote the root function obtained in Corollary 4.5. In the following, we prove that for any m ≥ k + 2, (5.1) φ -1

z i k (λ 0 ) n m-k+2 √ Le 2πiθ k (t) ∈ P m-1 λ 0 (v + λ 0 ). For each m ≥ k + 2, choose r m such that 1 < ρ k (r m ) ≤ n m-k+2 √ L and λ m = Φ -1 U (r m e 2πit ) ∈ P m+2 λ 0 ∩ R t U . By Corollary 4.5, v + λm = φ -1 z i k (λm) ρ k (r m )e 2πiθ k (t) . By Corollary 3.2, there is no critical values in f -m-1 λ X L λ \ {∞} for λ ∈ P m+1 λ 0 since no critical values of f λ 0 in f -m-1 λ 0 X L λ 0 \ {∞}. Hence φ -1 z i k (λ) n m-k+2 √ Le 2πiθ k (t) ∈ I m+2 λ is well-defined for λ ∈ P m+2 λ 0 . Since both v + λm = φ -1 z i k (λm) ρ k (r m )e 2πiθ k (t) and φ -1 z i k (λm) n m-k+2 √ Le 2πiθ k (t) belong to the dynamical ray R θ k (t) z i k (λm)
, and

I m-1 λ does not contain the equipotential curve E s z i k (λm) with 1 < s < n m-k+2 √ L, we have (5.2) φ -1 z i k (λm) n m-k+2 √ Le 2πiθ k (t) ∈ P m-1 λm (v + λm ).
By Corollary 3.2,

h m : P m λ 0 × ∂P m-1 λ 0 φ -1 z i k (λ 0 ) n m-k+2 √ Le 2πiθ k (t) ∪ v + λ 0 → P m λ 0 × C
is a holomorphic motion. Hence equation (5.2) implies that

φ -1 z i k (λ) n m-k+2 √ Le 2πiθ k (t) ∈ P m-1 λ v + λ
for all λ ∈ P m λ 0 . Then (5.1) follows by setting λ = λ 0 . Certainly, for 1 < s < n m-k+2 √ L, we also have

(5.3) φ -1 z i k (λ 0 ) se πiθ k (t) ∈ P m-1 λ 0 v + λ 0 . Let s → 1. Then w t k (λ 0 ) := lim s→1 φ -1 z i k (λ 0 ) se 2πiθ k (t) ∈ P m-1 λ 0 v + λ 0 for all m ≥ k + 2. This follows that w t k (λ 0 ) ∈ m≥k+2 P m-1 λ 0 v + λ 0 = K + λ 0 . For H 0 , λ = Φ -1 H 0 (e 2πit ) ∈ F 0 if and only if t ∈ 0, 1 n-1 and Φ -1 H 0 (1) ∈ ∂F 0 . Corollary 5.3. Suppose U ⊂ H k is an escape component. Then for λ = Φ -1 U (e 2πit )∩ F 0 , f k-1 λ K + λ ∩ ∂B λ = f k-1 λ w t k (λ) . Proof. By Proposition 5.2, f k-1 λ (w t k (λ)) ∈ f k-1 λ K + λ . Noting that f k-1 λ φ -1 z i k (λ 0 ) se 2πiθ k (t) ∈ B λ yields f k-1 λ (w t k (λ)) ∈ f k-1 λ K + λ ∩ ∂B λ . Conversely, by Lemma 5.1, f k-1 λ K + λ ∩ ∂B λ = n≥0 P n λ f k-1 λ (v + λ ) ∩ ∂B λ contains at most one point. This implies that f k-1 λ K + λ ∩ ∂B λ = f k-1 λ (w t k (λ)) .

No escape components attached on ∂H 0

The purpose of this section is to prove that the boundary of the Cantor locus is disjoint with the boundary of any escape component. This result is also the key step of proving Theorem 1.2. 6.1. Assumptions. Let U ⊂ H k be an escape hyperbolic component with order k ≥ 2. To prove that ∂U ∩∂H 0 = ∅ is impossible, it suffice to show that R t 0 H 0 ∩R t 1 U ∩ F 0 = ∅ for any t 0 , t 1 ∈ R/Z. We prove this by seeking a contradiction under the assumption

R t 0 H 0 ∩ R t 1 U ∩ F 0 = ∅. Let us use the notation in the proof of Proposition 5.2. z i k (λ) is the root function defined on U * ⊃ U , such that v + λ U = z i k (λ U ). Proposition 6.1. Let U ⊂ H k be an escape hyperbolic component with order k ≥ 2. If R t 0 H 0 ∩ R t 1 U ∩ F 0 = ∅, then τ k-1 (t 0 ) = t 0 .
Proof. Suppose λ 0 ∈ R t 0 H 0 ∩R t 1 U ∩F 0 . By Proposition 5.2, w t 0 0 (λ 0 ), w t 1 k (λ 0 ) ⊂ K + λ 0 . Then (6.1)

f k-2 λ 0 (w t 0 0 (λ 0 )), f k-2 λ 0 (w t 1 k (λ 0 )) ⊂ f k-2 λ 0 (K + λ 0 ) and f k-1 λ 0 (w t 0 0 (λ 0 )), f k-1 λ 0 (w t 1 k (λ 0 )) ⊂ f k-1 λ 0 (K + λ 0 ) ∩ ∂B λ . By Corollary 5.3, we get (6.2) f k-1 λ 0 (w t 1 k (λ 0 )) = f k-1 λ 0 (w t 0 0 (λ 0 )), If f k-2 λ 0 (w t 0 0 (λ 0 )) = f k-2 λ 0 (w t 1 k (λ 0 )
), then by (6.1), (6.2) and the fact

f k-2 λ 0 (K + λ 0 ) ⊂ P m λ 0 f k-2 λ 0 (w t 0 0 (λ 0
)) for all integers m ≥ 0, we have that

f λ 0 : P m+1 λ 0 f k-2 λ 0 (w t 0 0 (λ 0 )) → P m λ 0 f k-1 λ 0 (w t 0 0 (λ 0 )
) is a ramified covering of degree at least two. Since the puzzle pieces are all simply connected, P m λ 0 f k-1 λ 0 (w t 0 0 (λ 0 )) must contain a critical value for all m ≥ 0. It follows that f

k-1 λ 0 (K + λ 0 ) contains a critical value v + λ 0 or v - λ 0 , that is, f k-1 λ 0 (K + λ 0 ) = K + λ 0 or = K - λ 0 . This implies that τ k-1 t 0 2 = t 0 2 or τ k-1 t 0 2 = t 0 2 + 1 2 . Both lead to the consequence that τ k-1 (t 0 ) = t 0 . If f k-2 λ 0 (w t 0 0 (λ 0 )) = f k-2 λ 0 (w t 1 k (λ 0 )), then both dynamical rays f k-2 λ 0 R θ k (t 1 ) z i k (λ 0 ) ⊂ T λ 0 and f k-2 λ 0 R θ 0 (t 0 ) ∞(λ 0 )
⊂ B λ 0 land on the common point f k-2 λ 0 (w t 0 0 (λ 0 )). Note that

f k-1 λ 0 R θ k (t 1 ) z i k (λ 0 ) = f k-1 λ 0 R θ 0 (t 0 ) ∞(λ 0 ) ⊂ B λ 0 is the external ray landing on f k-1 λ 0 (w t 0 0 (λ 0 )). It follows that f k-2 λ 0 (w t 0 0 (λ 0 )) is a critical point and f k-1 λ 0 (w t 0 0 (λ 0 )) ∈ f k-1 λ 0 (K + λ 0 ) is a critical value v + λ 0 or v - λ 0 .
We again obtain that τ k-1 t 0 2 = t 0 2 or τ k-1 t 0 2 = t 0 2 + 1 2 , and then τ k-1 (t 0 ) = t 0 . Suppose that t 0 ∈ 0, 1 n-1 satisfying τ p t 0 = t 0 for an integer p ≥ 1. Let λ 0 = Φ -1 H 0 (e 2πit 0 ). It follows from [START_REF] Qiu | Hyperbolic components of McMullen maps[END_REF] (Lemma 4.6, Lemma 4.8, Theorem 6.2 and Remark 6.3) that there is a quadratic-like map g λ 0 : U → V with a parabolic fixed point β λ 0 ∈ U satisfying g λ 0 (β λ 0 ) = β λ 0 , g λ 0 (β λ 0 ) = 1, and g λ 0 (β λ 0 ) = 0 which is defined according to the following 3 cases.

(1) If

τ p t 0 2 = t 0 2 , then g λ 0 = f p λ 0 , U = P N λ 0 v + λ 0 and V = P N -p λ 0 v + λ 0 for an N large, β λ 0 = w t 0 0 (λ 0 ), and v + λ 0 is the unique critical value. (2) If τ p t 0 2 = t 0 2 + 1 2 and τ p t 0 2 + 1 2 = t 0 2 + 1 2 , then g λ 0 = f p λ 0 , U = P N λ 0 v - λ 0 and V = P N -p λ 0 v - λ 0 for an N large, β λ 0 = -w t 0 0 (λ 0 ), and v - λ 0 is the unique critical value. (3) If τ p t 0 2 = t 0 2 + 1 2 and τ p t 0 2 + 1 2 = t 0 2 , then g λ 0 = -f p λ 0 , U = P N λ 0 v + λ 0 and V = P N -p λ 0 v +
λ 0 for an N large, β λ 0 = w t 0 0 (λ 0 ), and v + λ 0 is the unique critical value.

Remark 6.1. In the above, we have assumed that λ 0 / ∈ R + . When λ 0 ∈ R + , the regions U, V will be taken those constructed in the proof of Lemma 7.2 in [START_REF] Qiu | Dynamics of McMullen maps[END_REF].

In the following discussion, without loss of generality, we may assume we are in the case (1). Hence we have the following Assumption 6.2. Assumption 6.2. Suppose that U ⊂ H k is an escape component with k ≥ 2, λ 0 ∈ R t 0 H 0 ∩ R t 1 U ∩ F 0 , and p|(k -1) is a positive integer such that τ p t 0 2 = t 0 2 . Then (1) there exists N such that g λ 0 = f p λ 0 :

P N λ 0 v + λ 0 → P N -p λ 0 v + λ 0 is a quadratic- like map with unique critical value v + λ 0 ; (2) g λ 0 w t 0 0 (λ 0 ) = w t 0 0 (λ 0 ), g λ 0 w t 0 0 (λ 0 ) = 1, and g λ 0 w t 0 0 (λ 0 ) = 0; (3) g (k-1)/p λ 0 w t 1 k (λ 0 ) = w t 0 0 (λ 0 ), where w t 0 0 (λ 0 ) is the landing point of ray R θ 0 (t 0 ) ∞(λ 0 ) and w t 1 k (λ 0 ) is the landing point of ray R θ k (t 1 ) z i
k (λ 0 ) . 6.2. Rational Family with Parabolic implosion. The original parabolic implosion theory is invented by A. Douady and P. Lavaurs [START_REF] Lavaurs | Systemes dynamiques holomorphes: explosion de points périodiques paraboliques[END_REF], see [START_REF] Douady | Etude dynamique des polynômes complexes[END_REF]. This theory is further developed by M. Shishikura , see [START_REF] Shishikura | The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets[END_REF][START_REF] Shishikura | Bifurcation of parabolic fixed points[END_REF]. Here we use the terminologies and results given in [START_REF] Shishikura | Bifurcation of parabolic fixed points[END_REF][START_REF] Lei | Local properties of the Mandelbrot set at parabolic points[END_REF].

Let ∆ ρ,θ be a bounded connected open set with 0 on its boundary, and F : ∆ ρ,θ × C → C, (u, z) → F u (z) be a map satisfying the following conditions.

(1) F : (u, z) → F u (z) is continuous on ∆ ρ,θ × C, holomorphic on ∆ ρ,θ × C, and F u (z) is a rational map for each u ∈ ∆ ρ,θ .

(2) F u (z) = m(u)z + O(z 2 ) as z → 0 with m(0) = 1, F 0 (0) = 0, and F 0 has a unique simple critical value v 0 contained in the parabolic basin of 0.

(3) σ(u) := m(u) -1 2πi maps ∆ ρ,θ univalently onto {z ∈ C : |z| < ρ, | arg z| < θ} with ρ > 0 small and θ ∈ (0, π). By (2), F u has two fixed points 0 and q(u) when u ∈ ∆ ρ,θ \ {0}. The parabolic implosion phenomenon for such a family F u was given as the Douday-Lavaurs-Shishikura theorem (see Theorem 2.1 in [START_REF] Lei | Local properties of the Mandelbrot set at parabolic points[END_REF]). Since we need only to use the attracting part of Theorem 2.1 in [START_REF] Lei | Local properties of the Mandelbrot set at parabolic points[END_REF], we translate it to the following Theorem 6.3. Set S M = {z ∈ C : 0 < (z) < M + 1} for any M ∈ N. Theorem 6.3 (Douady-Lavaurs-Shishikura). Let F : ∆ ρ,θ × C → C, (u, z) → F u (z) satisfy conditions given above. For any given M ∈ N large, when ρ is small enough, there exists a continuous map Ψ : ∆ ρ,θ ×S M → C, (u, z) → Ψ u (z) satisfying following properties (see Figure 7).

(1) For u = 0, Ψ 0 : S M → Ψ 0 (S M ) ⊂ Ω 0 is a univalent map where Ω 0 is the attracting petal of F 0 with the parabolic fixed point 0 on its boundary, and

Ψ -1 0 is the restriction of the usual Fatou coordinate ϕ 0 : Ω 0 → {z : (z) > 0}. (2) For u ∈ ∆ ρ,θ \ {0}, Ψ u : S M → C is a univalent map such that Ψ u (S M ) is a
Jordan region containing two fixed points 0 and q(u) of F u on its boundary.

(3) Whenever z, z + 1 ∈ S M , Ψ u (z + 1) = F u (Ψ u (z)). Furthermore, Ψ u (S 0 ) contains the critical value v(u) of F u with v(0) = v 0 .
By this theorem, we call the map F or the family F u the rational family with parabolic implosion (RFPI). Remark 6.2. From (3), Ψ u (S 0 ) is a fundamental region of F u , and F m u is well-defined and univalent on Ψ u (S 0 ) for any 1 ≤ m ≤ M . 6.3. Application to McMullen family. Set g λ := f p λ , where p is given in Assumption 6.2. Consider the equation g λ (z)z = 0. By part (2) of Assumption 6.2, the fixed points of g λ (z) for (λ, z) near λ 0 , w t 0 0 (λ 0 ) can be uniformized by setting λ = λ 0 + u 2 . That is, there exist r > 0 small enough such that D(λ 0 , r) ⊂ U * (U * is defined in Proposition 5.2), a local change of coordinates λ : D(0, √ r) → D(λ 0 , r) given by u → λ 0 + u 2 , and a holomorphic function p : D(0, √ r) → C such that p(u) and p(-u) are the two local fixed points of g λ 0 +u 2 satisfying p(u) = w t 0 0 (λ 0 + u 2 ) for

λ 0 + u 2 ∈ D(0, √ r) \ H 0 . Let m : D(0, √ r) → C, u → g λ 0 +u 2 (p(u)
) be the multiplier of the fixed point p(u). Then m(u) is holomorphic on D(0, √ r) and m(0) = 1. It is evident that

(6.3) g λ 0 +u 2 (z) = p(u) + m(u)(z -p(u)) + O(z -p(u)) 2 as z → p(u). Let T u (z) := z -p(u) and F u (ω) := T u • g λ 0 +u 2 • T -1 u (ω)
. Then (6.3) becomes (6.4)

F u (ω) = m(u)ω + O(ω 2 ) as ω → 0. 1 2 3 4 -1 -2 1 2 S1 \ S0 v(0) v(0) v(u) v(u) Ψu Ψu Ψ0 Ψ0 F0 Fu T (x) = x + 1 S0 1 2 3 4 -1 -2 1 2 Figure 7
. illustration of Theorem 6.3 Remark 6.3. It is obvious that other fixed point p(-u) of g λ 0 +u 2 has the multiplier m(-u). So except the fixed point 0, the other fixed point of F u is q(u) = p(-u)-p(u) with the multiplier m(-u) when u = 0.

Let v(u) := T u (v + λ 0 +u 2 ) which is the critical value of F u (ω). Let U 0 = T 0 (P N λ 0 (v + λ 0 )) and V 0 := T 0 (P N -p λ 0 (v + λ 0 )). Then by (1) of Assumption 6.2, F 0 : U 0 → V 0 is a quadratic-like map with the unique critical value v(0) ∈ U 0 which is obviously contained in the parabolic basin of 0. When r > 0 is small enough, then by the continuity of F u with respect to u ∈ D(0, √ r), it is easy to find a simply-connected region

U u ⊂ U 0 such that F u : U u → V u := F u (U u ) is also a quadratic-like map with the unique critical value v(u) ∈ U u . Furthermore U u → U 0 as u → 0 in Hausdorff topology. Let Γ ± := λ -1 R t 1 U
denote the two pre-images of R t 1 U in the u-plane which can be parameterized by Γ ± (s) := λ -1 (Φ U (se 2πit 1 )). Obviously, Γ ± (s) → 0 as s → 1. Proposition 6.4. For any θ ∈ (0, π/2), there exist ρ > 0 and

∆ ρ,θ ⊂ D(0, √ r) which is mapped univalently onto {z ∈ C : |z| < ρ, | arg z| < θ} by σ = (m -1)/(2πi) such that Γ + (s) ∈ ∆ ρ,θ or Γ -(s) ∈ ∆ ρ,θ for s sufficiently close to 1.
The proof of Proposition 6.4 will be given in the next subsection. By Proposition 6.4 and the discussion above,

F : ∆ ρ,θ × C → C defined by F (u, z) := F u (z) is a RFPI. 1 2 3 4 5 6 -1 -2 1 2 S0 1 2 3 4 5 6 -1 -2 1 2 SM \ SM-1 v(u) Ψu Ψu Fu T (x) = x + k-1 p v(u) Ru Ru Figure 8
. illustration for the proof of Proposition 6.5 Proposition 6.5. For any escape component

U ⊂ H k with k ≥ 2, U ∩ H 0 = ∅.
Proof. If the Proposition is not true, we may assume Assumption 6.2 stands. By Proposition 6.4, without loss of generality, let us assume Γ + (s

) ∈ ∆ ρ,θ for s(> 1) close to 1. Let R u := T u R θ 0 (t 0 ) ∞(λ 0 ) parameterized by R u (s) := T u φ -1 ∞(λ) se 2πiθ 0 (t 0 ) , Ru := T u R θ k (t 1 ) z i k (λ 0 +u 2 ) parameterized by Ru (s) := T u φ -1 z i k (λ 0 +u 2 ) se 2πiθ k (t 1 ) . Then we have F u (R u ) = R u and F (k-1)/p u ( Ru ) = R u by (3) of Assumption 6.2. Note that by Corollary 4.5, the critical value v(u) = Ru (ρ k (s)) for u = Γ + (s) ∈ ∆ ρ,θ close to 0. Hence F (k-1)/p u (v(u)) = R u (s 0 ) ∈ R u for some s 0 > 1. Take M = (k -1)/p and Ψ : ∆ ρ,θ × S M → C as given in Theorem 6.3. Then by part (3) of Theorem 6.3 (see Remark 6.2), v(u) ∈ Ψ u (S 0 ) and R u (s 0 ) = F M u (v(u)) ∈ Ψ u (S M ). On the other hand, in Remark 2.1, it is pointed that R u can be extended con- tinuously to R u n M √ s 0 such that F M u R u n M √ s 0 = R u (s 0 ). The invariance and continuity of R u ensures that R u n M √ s 0 ∈ Ψ u (S 0 ). By Remark 6.2, F M u is univalent on Ψ u (S 0 ). Hence, F M u (v(u)) = F M u R u ( n M √ s 0 ) = R u (s 0 ) implies that v(u) = R u n M √ s 0 ∈ R u . This contradicts with v(u) = Ru (ρ k (s)) ∈ Ru . 6.4. Proof of Proposition 6.4. Let X ρ,θ := |z -1| < 2πρ : | arg(z -1)| ≤ π 2 -θ and Y ± ρ,θ := {z ∈ H ± : |z -1| < 2πρ} \ (-X ρ,θ ∪ X ρ,θ
(z) = z -p(u), g λ 0 +u 2 is conjugated to F u (ω) = m(u)ω + O ω 2 . Since the multiplier m(u) is a non-constant holomorphic function in D(0, √ r) and m(0) = 1, m(u) has the following expansion (6.5) m(u) = 1 + au + o(u ),
where is a positive integer and a = 0.

Lemma 6.6. is an odd number.

Proof. It is known that for u ∈ D(0, √ r) with r > 0 small, F u : U 0 → V u := F u (U 0 ) is a quadratic-like map with the unique critical value v(u) ∈ U 0 . From Remark 6.3, F u has two fixed pionts 0 and q(u) = p(-u)p(u) contained in U 0 with the multipliers m(u) and m(-u) respectively. Suppose is even, then from (6.5), we have

m(u) -m(-u) = o(u ), which implies that |m(u) -m(-u)| < |m(u) -1| for u small since m(u) -1 ∼ au . It follows that |m(-u)| ≤ |m(u) -m(-u)| + |m(u)| < |1 -m(u)| + |m(u)| = 1 -|m(u)| + |m(u)| = 1, (6.6)
Hence q(u) = p(-u)p(u) is an attracting fixed point of F u . On the other hand, by the open mapping theorem, we can find u close to 0 and q(u) ∈ U u such that |m(u)| < 1. So 0 is also an attracting fixed point of F u for such u. This is impossible since F u is a quadratic-like map on U u .

0.5 1.0 1.5 2.0 2.5 3.0 -0.5 -1.0 -0.5 -1.0 0.5 1.0 X ρ,θ Y + ρ,θ m(u) r(u) θ(u) 1 -r(u) D Figure 9
. illustration for the proof of Lemma 6.7 Lemma 6.7. For any θ ∈ 0, π 2 , if u is small and

| arg(m(u) -1)| < π 2 -θ, then m(-u) ∈ D.
Proof. By (6.5) and Lemma 6.6, m(u) + m(-u) -2 = o(u ), and then (6.7)

|m(u) + m(-u) -2| = o (|m(u) -1|)
for u small. Let r(u) := |m(u) -2| and θ(u) := arg(m(u) -1). It follows that for u small,

|m(u) -1| = cos(θ(u)) -(r(u)) 2 -sin 2 (θ(u)) = cos(θ(u)) -(r(u) -1) 2 + 2(r(u) -1) + cos 2 (θ(u)) = cos(θ(u)) -cos(θ(u)) 1 + 2r(u) -2 + (r(u) -1) 2 cos 2 (θ(u)) = cos(θ(u)) -cos(θ(u)) 1 + r(u) -1 cos 2 (θ(u)) + o(r(u) -1) = 1 -r(u) cos(θ(u)) + o(r(u) -1).
Hence for u small and |θ(u

)| ≤ π 2 -θ < π 2 , we have (6.8) |m(u) -1| = O(1 -|2 -m(u)|).
Coupling it with (6.7) gives that |m(u)

+ m(-u) -2| < 1 -|2 -m(u)| for u small. Hence |m(-u)| ≤ |m(u) + m(-u) -2| + |m(u) -2| < 1 -|2 -m(u)| + |m(u) -2| = 1. (6.9)
Proposition 6.8. For any θ ∈ 0, π 2 and ρ > 0, one of the following is true.

• m(Γ + (s)) ∈ Y + ρ,θ for all s > 1 sufficiently close to 1; • m(Γ -(s)) ∈ Y +
ρ,θ for all s > 1 sufficiently close to 1.

Proof. Fix any θ ∈ 0, π 2 and ρ > 0. Since Γ ± (s) → 0 and m(Γ ± (s)) → 1 as s → 1, we have |m(Γ ± (s)) -1| < ρ for s > 1 sufficiently close to 1. Since f λ 0 +u 2 is an escape map for u = Γ ± (s), F u has no attracting fixed points in C. Hence, it follows from Lemma 6.7, m(Γ It follows that (6.12)

± (s)) ∈ Y + ρ,θ ∪ Y - ρ,θ for all s close to 1. Assume that m(Γ -(s)) ∈ Y - ρ,
(m(Γ + (s))) = (2 -m(Γ -(s))) -(2 -m(Γ -(s)) -m(Γ + (s))) = (1 -m(Γ -(s))) -(2 -m(Γ -(t)) -m(Γ + (s))) > (1 -m(Γ -(s))) -|2 -m(Γ -(s)) -m(Γ + (s))| > (1 -m(Γ -(s))) -(1 -m(Γ -(s))) = 0.
Hence m(Γ + (s)) ∈ Y + ρ,θ for all s > 1 close to 1. Proof of Proposition 6.4. Take ρ small enough. By Proposition 6.8, we can choose an inverse branch m -1 of m such that m -1 (Y + ρ,θ ) contains Γ + (s) or Γ -for s close to 1. Let ∆ ρ,θ := m -1 Y + ρ,θ ⊂ D(0, √ r). Then m : ∆ ρ,θ → Y + ρ,θ is a univalent map. According to the relation of σ and m, the result follows directly.

Proof of the main theorems

In this section, we will present the proof of Theorem 1.2 and Theorem 1.1. 7.1. Closures of escape components are pairwise disjoint. In this subsection, we finish the proof of Theorem 1.2. By Proposition 6.5, it remains to show that the closures of escape components with order at least 2 are pairwise disjoint. Then the result follows easily by Proposition 6.8. Lemma 7.1. There is no parameter λ ∈ ∂H such that

f m λ (v + λ ) ∈ C λ with m ≥ 0. Proof. Suppose if λ satisfies f m λ (v + λ ) ∈ C λ with m ≥ 0. Then either f m+1 λ v + λ = v + λ or f m+1 λ v + λ = v - λ .
In the first case, we get f m λ v + λ is a periodic critical point, which is clearly impossible.

In the second case, recall the McMullen map satisfies f m+1 λ

(-z) = -f m+1 λ (z) or f m+1 λ (-z) = f m+1 λ (z) depend on n is odd or even. So if n is odd, then f m+1 λ (v - λ ) = -f m+1 λ (v + λ ) = v + λ . This means f 2m+2 λ v + λ = v + λ , so deduce to the first case. If n is even, then f m+1 λ (v - λ ) = f m+1 λ (v + λ ) = v - λ . So f m λ (v - λ ) is a periodic m + 1 critical point, which is also clearly impossible. Proposition 7.2. Suppose U 1 ⊂ H k 1 and U 2 ⊂ H k 2 are two distinct escape com- ponents with order k 1 , k 2 ≥ 2, then ∂U 1 ∩ ∂U 2 = ∅. Proof. Suppose λ 0 = R t 1 U 1 ∩ R t 2 U 2 ∈ ∂U 1 ∩ ∂U 2 .
Without loss of generality, we may assume that λ 0 ∈ F 0 . Let V j be a simply-connected open subset of C * \ ≤k j -1 Λ containing λ 0 and λ U j for j = 1, 2. By Lemma 4.1, there exist two root functions z i j k j (λ) defined on V j for j = 1, 2 such that v + λ U j = z i j k j (λ U j ). Then both z i 1 k 1 (λ 0 ) and z i 2 k 2 (λ 0 ) are well defined. Let U i j k j (λ 0 ) denote the Fatou component containing z i j k j (λ 0 ) and let

w t j k j (λ 0 ) := lim s→1 φ -1 z i j k j (λ 0 )
se 2πi θ k j (t j )

for j = 1, 2. That is, w

t j k j (λ 0 ) is the landing point of dynamical ray R j := R t j z i j k j (λ 0 ) in U i j k j (λ 0 ). By Proposition 5.2, w t 1 k 1 (λ 0 ), w t 2 k 2 (λ 0 ) ⊂ K + λ 0 . By part 2 of Theorem 2.6, K + λ 0 = v + λ 0 . It follows that w t 1 k 1 (λ 0 ) = w t 2 k 2 (λ 0 ) = v + λ 0 , which implies the dynamical rays R 1 and R 2 land together at the common point v + λ 0 ∈ ∂U i 1 k 1 (λ 0 ) ∩ ∂U i 2 k 2 (λ 0 ). Suppose z i 1 k 1 (λ 0 ) = z i 2 k 2 (λ 0
). Notice that both U i 1 k 1 (λ 0 ) and U i 2 k 2 (λ 0 ) will be mapped eventually to the Fatou component B λ 0 and B λ 0 is invariant. We get that there exists m > 0 such that

f m λ 0 (U i 1 k 1 (λ 0 )) = f m λ 0 (U i 2 k 2 (λ 0 )) but f m+1 λ 0 (U i 1 k 1 (λ 0 )) = f m+1 λ 0 (U i 2 k 2 (λ 0 )). (In fact, if k 1 = k 2 , then m = max{k 1 , k 2 } -2; if k 1 = k 2 , then m < k 1 -2.) Thus, f m λ 0 (R 1 ) and f m λ 0 (R 2 ) are two different dynamical rays which land together at f m λ 0 (v + λ 0 ), but f m+1 λ 0 (R 1 ) = f m+1 λ 0 (R 2 ) is a dynamical ray landing on f m+1 λ 0 (v + λ 0 ) since the Fatou component f m+1 λ 0 (U i 1 k 1 ) = f m+1 λ 0 (U i 2 k 2
) is a Jordan domain (see part (1) of Theroem 2.6) and there is only one dynamical ray landing on f m+1 λ 0 (v + λ 0 ). This implies f m λ 0 (v + λ 0 ) ∈ C λ 0 which contradicts with Lemma 7.1. Suppose

z i 1 k 1 (λ 0 ) = z i 2 k 2 (λ 0 ). In this case k 1 = k 2 ≥ 3 and U i 1 k 1 (λ 0 ) = U i 2 k 2 (λ 0 ). Since ∂U i 1 k 1 (λ 0
) is a Jordan curve and w t 1 k 1 (λ 0 ) = w t 2 k 2 (λ 0 ), we have t 1 = t 2 . By the discreteness of pre-images of 0, it is not hard to see that

z i 1 k 1 (λ) = z i 2 k 2 (λ) for λ near λ 0 . Hence we can find a simply-connected region V ⊂ V 1 ∪ V 2 containing λ U 1 , λ U 2 and λ 0 such that z i 1 k 1 (λ) = z i 2 k 2 (λ) for λ ∈ V. By Proposition 4.4, we have λ U 2 = e 2mπi n-1 λ U 1 for some m ∈ N. It follows that R t 1 U 1 ∪ R t 2 U 2 ∩ R + = ∅.
Without loss of generality, we may suppose R t 1 U 1 ∩ R + = ∅ and λ ∈ R t 1 U 1 ∩ R + . It is not hard to check that for λ ∈ R + , both B λ ∩ R + and T λ ∩ R + are connected. Hence we may suppose

B λ ∩ R + = (z 0 , ∞), T λ ∩ R + = (0, z 1 ). Since λ ∈ R t 1 U 1 , then v + λ ∈ [z 1 , z 0 ] and f k 1 λ v + λ ∈ [0, z 1 ]. Notice that v + λ = min z∈R + f λ (z). It follows that λ = λ 0 v + λ = z 1 ∈ ∂T λ ∩ ∂U i 1 k 1 (λ )
which can be deduced to previous case. A combination of Proposition 6.5 and Proposition 7.2 completes the proof of Theorem 1.2. The proof of Proposition 7.2 also implies the following.

Corollary 7.3. Suppose U ⊂ H k is an escape component with k ≥ 2. If λ = Φ -1 U (e 2πit ) ∈ ∂U , then v + λ = φ -1 z i k (λ) e 2πiθ k (t) ∈ ∂U i k (λ), where z i k is the root function defined on a neighborhood of U satisfying that v + λ U = z i k (λ U ) and U i k (λ) is the Fatou component containing z i k (λ). In particular v ± λ / ∈ ∂B λ . 7
H : W × B λ 0 → W × C. Proof. When λ ∈ W \ {0}, C λ ∩ B λ = ∅ and B λ is a Jordan region. The Böttcher coordinate φ ∞(λ) is a holomorphic homeomorphism from B λ to C \ D. When λ = 0, set B 0 = C\D and φ ∞(0) = id.
By the usual construction of the Böttcher coordinates, it is easy to check φ ∞(λ) is holomorphic with respect to λ ∈ W , even at λ = 0. Hence

H λ (z) := φ -1 ∞(λ) • φ ∞(λ 0 ) (z), z ∈ B λ 0 is well-defined for λ ∈ W . Then it is direct to verify that H : W × B λ 0 → W × C, (λ, z) → (λ, H λ (z)) is a holomorphic motion.
By Slodkowski's Theorem 2.3, the holomorphic motion defined in Lemma 7.4 can be extended to a holomorphic motion

H : W × C → W × C. It follows that the following map (7.2) Φ k (λ) := H -1 λ (f k-1 λ (v + λ )), λ ∈ W \ {0}, ∞, λ = 0 is well-defined on W for all k ≥ 2. Noting that when λ → 0, H -1 λ (f k-1 λ (v + λ )) → ∞, Φ k (λ) is continuous even at λ = 0.
In the following, we always assume that λ 0 ∈ W is given.

Lemma 7.5. The map Φ k : W → C defined by (7.2) is quasi-regular on any region W * W .

Proof. Consider the derivative of equation

H λ • Φ k (λ) = f k-1 λ (v + λ ). Since H λ and f k-1 λ (v + λ ) are all holomorphic in λ. Therefore, ∂H λ /∂λ = ∂f k-1 λ (v + λ )/∂λ = 0, we have (7.3) ∂H λ ∂z Φ k (λ) ∂Φ k ∂λ λ + ∂H λ ∂z Φ k (λ) ∂Φ k ∂λ λ = 0,
where ∂H λ /∂z and ∂H λ /∂z exist almost everywhere since H λ is quasi-conformal. Thus (7.4)

∂Φ k /∂λ ∂Φ k /∂λ λ = ∂Φ k /∂λ ∂Φ k /∂λ λ = ∂H λ /∂z ∂H λ /∂z Φ k (λ) = |µ λ (Φ k (λ))| , where µ λ is the Beltrami coefficient of H λ . Let ρ(•, •) denote the hyperbolic distance of W . Then ρ * = sup λ∈W * ρ(λ, λ 0 ) < ∞.
By Theorem 2.3, for any λ ∈ W * , (7.5) ess sup

z∈C |µ λ (z)| ≤ e ρ(λ,λ 0 ) -1 e ρ(λ,λ 0 ) + 1 ≤ k := e ρ * -1 e ρ * + 1 < 1. Therefore (7.6) ∂Φ k /∂λ ∂Φ k /∂λ ∞ = ess sup λ∈W * ∂Φ k /∂λ ∂Φ k /∂λ = ess sup λ∈W * |µ λ (Φ k (λ))| ≤ k < 1.
It means that Φ k is a quasi-regular map on the region W * .

From the proof of Lemma 7.5, we have

Corollary 7.6. Let µ Φ k (λ) = (∂Φ k /∂λ)/(∂Φ k /∂λ) be the Beltrami coefficient of Φ k (λ). Let D m = D(λ 0 , 1/m) for m > 0 large such that D m ⊂ W . Then (7.7) µ m,k := ess sup λ∈Dm |µ Φ k (λ)| → 0 as m → ∞.
Proof. It comes immediately from (7.5), (7.6) and the fact that sup λ∈Dm ρ(λ, λ 0 ) → 0 as m → ∞.

Proposition 7.7. Let U be an escape component of H k with k ≥ 2. Then there is a neighborhood V of U and a quasi-conformal homeomorphism

Ψ k : V → C such that Ψ k (U ) = B λ 0 . Proof. Set Ψ k : W → C defined by (7.8) Ψ k :=    φ -1 ∞(λ 0 ) • φ ∞(λ 0 ) • Φ k 2/(n-2) , k = 2, φ -1 ∞(λ 0 ) • φ ∞(λ 0 ) • Φ k 1/n , k ≥ 3,
or more clearly, using (7.2), Ψ k (0) = ∞ and for λ = 0

(7.9) Ψ k (λ) = φ -1 ∞(λ 0 ) • φ ∞(λ 0 ) • H -1 λ (f λ (v + λ )) 2/(n-2) , k = 2, φ -1 ∞(λ 0 ) • φ ∞(λ 0 ) • H -1 λ (f k-1 λ (v + λ )) 1/n , k ≥ 3. When λ ∈ U , it is easy to check that Ψ k (λ) = φ -1 ∞(λ 0 ) • Φ U (λ)
, where Φ U (λ) is defined in Theorem 2.2 which is a holomorphic homeomorphism from U to C \ D (here for k = 2, we need extend the definition of Φ U such that Φ U (0) = ∞). It follows that Ψ k : U → B λ 0 is a holomorphic homeomorphism. By parts (1), (3) of Theorem 2.6, both ∂U and ∂B λ 0 are Jordan curves. Hence Ψ k : U → B λ 0 is a homeomorphism with Ψ k (∂U ) = ∂B λ 0 . By Proposition 6.5 and Proposition 7.2, we have U ∩ U = ∅ for any escape component U different from U . Since for fixed ≤ k, H has only finitely many components, there exists a simply-connected region W * such that U W * W and W * ∩ U = ∅ for all components U of H with 0 ≤ ≤ k which are different from U . Now, we restrict Ψ k on W * and then show that Ψ -1

k B λ 0 = U . That is, if λ ∈ W * such that Ψ k (λ) ∈ B λ 0 , then λ ∈ U .
We first prove that Ψ k (λ) ∈ B λ 0 implies λ ∈ U . From (7.9) and the definition of φ ∞(λ) , we have that Ψ k (λ) ∈ B λ 0 implies H -1 λ (f k-1 λ (v + λ )) ∈ B λ 0 . Since H λ : C → C is a quasi-conformal homeomorphism and H λ (B λ 0 ) = B λ , we get that f k-1 λ (v + λ ) ∈ B λ . This shows that λ ∈ H for some ≤ k. However, by the definition of W * , W * ∩ ≤k H = U . We get that λ ∈ U .

Suppose that Ψ k (λ) ∈ ∂B λ 0 . Since Φ k is quasi-regular by Lemma 7.5, Ψ k is also quasi-regular and obviously non-constant. Thus, Ψ k is an open map. It follows that for any neighborhood N of λ, Ψ k (N ) ∩ B λ 0 = ∅. Hence N ∩ U = ∅, which implies that λ ∈ ∂U .

Finally, we take a Jordan region V satisfying U V W * . Let γ = ∂V and Γ = Ψ k (∂V ). Then by discussion above, Γ ∩ B λ 0 = ∅. Let B be the component of C \ Γ which contains B λ 0 . Noting that a quasi-regular map is the composition of a holomorphic map and a quasi-conformal homeomorphism, the argument principle can be applied. Since for z ∈ B λ 0 ⊂ B has only one pre-image of Ψ k in U ⊂ V , we get that every z ∈ B has only one pre-image in V . Take V = Ψ k (U ) ⊂ V . Then V is a neighborhood of U and Ψ k : V → Ψ k (V ) = B is a quasi-conformal homeomorphism.

Corollary 7.8. ∂U is a quasi-circle.

Proof. Take λ 0 = λ U the center of U . Then f λ 0 is hyperbolic and hence λ 0 satisfies the condition of part (1) of Theorem 2.6. Hence ∂B λ 0 is a quasi-circle. By Proposition 7.7, ∂U is also a quasi-circle. Lemma 7.10. If f is a non-constant holomorphic map defined on a neighborhood of X ⊂ C, then dim H f (X) = dim H X.

Recall that for a K-quasi-conformal homeomorphism, we have the following Mori's theorem, see [START_REF] Ahlfors | Lectures on quasiconformal mappings[END_REF].

Theorem 7.11 (Mori). Suppose f : D → D is a K-quasi-conformal homeomorphism, then for each z 1 , z 2 ∈ D,

(7.10) |f (z 1 ) -f (z 2 )| ≤ 16|z 1 -z 2 | 1 K .
We also need the following theorem due to Przytycki [START_REF] Przytycki | On the hyperbolic Hausdorff dimension of the boundary of a basin of attraction for a holomorphic map and of quasirepellers[END_REF] as we known.

Theorem 7.12 (Przytycki). Let f : C → C be a rational map of degree d ≥ 2

and Ω be a simply-connected immediate basin of attraction to a periodic attracting point. Then provided f is not a Blaschke product in some holomorphic coordinates, or a quotient of a Blaschke product by a rational function of degree 2, the Hausdorff dimension of ∂Ω is greater than 1.

Corollary 7.13. Let U ⊂ H k be an escape component with order k ≥ 2. If λ 0 ∈ ∂U and z 0 ∈ ∂B λ 0 , then for any neighborhood U of z 0 , dim H (U ∩ ∂B λ 0 ) = dim H ∂B λ 0 > 1.

Proof. From Theorem 7.12, we have that dim H ∂B λ 0 > 1 as long as λ 0 ∈ W \ {0}.

So dim H ∂B λ 0 > 1 if λ 0 ∈ ∂U . For any neighborhood U of z 0 ∈ ∂B λ 0 , f m λ 0 (U ∩ ∂B λ 0 ) = ∂B λ 0 as m sufficiently large. By Lemma 7.10, dim H (U ∩ ∂B λ 0 ) = dim H f m λ 0 (U ∩ ∂B λ 0 ) = dim H ∂B λ 0 > 1.

Proposition 7.14. Let U ⊂ H k be an escape component with order k ≥ 2, then the Hausdorff dimension of ∂U satisfies Proof of Theorem 1.1. The proof is a combination of Corollary 7.8 and Proposition 7.14.

  1. Introduction 1.1. Main Reuslt. In this article, we consider the McMullen maps :

Theorem 1. 2 .

 2 The closures of escape components are pairwise disjoint.

Figure 1 .

 1 Figure 1. Parameter plane for n = 3.

Figure 2 .

 2 Figure 2. Parameter plane for n = 4.

  Overview of McMullen family. In 1988, C. McMullen [McM88] introduced a rational map f λ (z) = z 2 + λz -3 as a singular perturbation of z → z 2 . In his article, C. McMullen showed that for λ sufficiently small, the Julia set J(f λ ) of f λ is a Cantor circle (homeomorphic to a Cantor set × circle). In 2005, R. Devaney and his group [DL05, DLU05, BDL + 05] generalized the work of McMullen and studied a more general family of McMullen maps:

Figure 3 .

 3 Figure 3. For λ ∈ H 0 in Cantor locus, J(f λ ) is a Cantor set.

Figure 4 .

 4 Figure 4. For λ ∈ H 2 in Cantor locus, J(f λ ) is a Cantor circle.

Figure 5 .

 5 Figure 5. For λ in a Sierpinski hole, J(f λ ) is a Sierpinski carpet.

Figure 6 .

 6 Figure 6. For f λ is non-escape hyperbolic map, J(f λ ) is connected.

  θ for an s > 1, then by continuity, m(Γ -(s)) ∈ Y - ρ,θ for all s > 1 close to 1. It follows that (6.10) (1m(Γ -(s))) ≥ |1m(Γ -(s))| cos θ, Since Γ + (s) = -Γ -(s), we have from (6.7) and (6.10) that (6.11) m(Γ + (s)) + m(Γ -(s)) -2 = o( (1m(Γ -(s))).

7. 3 .

 3 Hausdorff dimension of boundary of escape component. Let dim H X denote the Hausdorff dimension of a Borel subset X of C. The following results are well-known, see[START_REF] Falconer | Fractal geometry: mathematical foundations and applications[END_REF].Lemma 7.9. Let X ⊂ C be a Borel subset. If f : X → C satisfies the Hölder condition |f (z 1 )f (z 2 )| ≤ C|z 1z 2 | α , then dim H f (X) ≤ 1 α dim H X.

  dim H ∂U < 2.Proof. Astala[START_REF] Astala | Area distortion of quasiconformal mappings[END_REF] proved that the image of a set of Hausdorff dimension 1 under a K-quasi-conformal homeomorphism has the Hausdorff dimension at most 1 + k, where k = (K -1)/(K + 1) < 1. It follows that any quasi-circle has the Hausdorff dimension less than 2. So dim H ∂U < 2 since ∂U is a quasi-circle by Corollary 7.8. It remains to show that dim H ∂U > 1.Choose a λ 0 ∈ ∂U and let D m = D(λ 0 , 1/m) for m > 0 large such that D m ⊂ V , where V is given in Proposition 7.7. By Corollary 7.6, (7.7) holds, i.e.µ m,k := ess sup λ∈Dm |µ Φ k (λ)| → 0 as m → ∞.Then, Φ k , as a quasi-regular map restricted on D m , has its maximal dilatation (7.12)K m,k := 1 + µ k,m 1µ k,m → 1 as m → ∞.Let Ψ k be the quasi-conformal homeomorphism defined in (7.8), let η m,k be the Riemann map from Ψ k (D m ) onto D, and let ξ m (λ) : D m → D be the affine map defined byξ m (λ) = m(λ-λ 0 ). Then Ψ m,k : D → D defined by Ψ m,k := η m,k •Ψ k •ξ -1m is a quasi-conformal homeomorphism. It has the maximal dilatation K m same as one of Φ k since φ ∞(λ 0 ) , η m,k and ξ m are all conformal. By Theorem 7.11 and Lemmas 7.9, 7.10,(7.13) dim H (∂U ∩ D m ) ≥ 1 K m dim H (Ψ k (∂U ∩ D m )) .By Corollary 7.13, there exists a constant c > 1 such thatdim H (Ψ k (∂U ∩ D m )) = dim H (U m,k ∩ ∂B λ 0 ) = dim H ∂B λ 0 ≥ c > 1, where U m,k = Ψ k (D m ) be a neighborhood of z 0 := Ψ k (λ 0 ) ∈ ∂B λ 0 . Since K m → 1 as m → ∞,then dim H (∂U ∩ D m ) > 1 for m large enough. Hence dim H ∂U ≥ dim H (∂U ∩ D m ) > 1.

  ). By part (3) of the definition of RFPI, it suffices to show that either m (Γ + (s)) ∈ Y + ρ,θ or m (Γ -(s)) ∈ Y + ρ,θ for s large. Recall that by the conjugacy under affine transform T u

  .2. McMullen domain and Sierpiński holes are quasi-disks. In this section, we prove that the McMullen domain and all Sierpiński holes are all bounded by quasi-circles. Recall that U is a McMullen domain or a Sierpiǹski hole if and only if U ⊂ H k is an escape component with k ≥ 2.

	Lemma 7.4. Let W = C \ H 0 . Then for any λ 0 ∈ W , there is a holomorphic motion based at λ 0
	(7.1)


Parametric study and energy analysis of a personalized ventilation system I. Cruceanu, C. Maalouf, I. Colda, M. Lachi Abstract: The personalized ventilation system aims to improve the quality of the inhaled air, to decrease the risk of transmission of the infectious agents and to improve the thermal comfort of the occupants.

In this work, a classical ventilation system is compared to a personalized ventilation system from the energy consumption and the thermal comfort point of view. The analysis is made with a nodal model, using the numerical simulations in the SPARK simulation environment. The object of the simulations is a classroom in Bucharest, Romania. Our results suggest that personalized ventilation can reduce the energy consumption of about 60%. Results are completed with a parametric study that shows that system performance is mainly affected by outdoor temperature variations and glazing solar heat gain coefficients which affects entering solar radiation creating internal heat loads.

Keywords: Personalized Ventilation, Classical Ventilation;

Energy Consumption; thermal comfort, simulation; SPARK.

I INTRODUCTION

The concept of personalized ventilation was first introduced by Fanger [START_REF] Po | Human requirements in future air-conditioned environements[END_REF] who said that "we would hesitate to drink water from a swimming pool polluted by human bio effluents. Still we accept consuming indoor air that has previously been in the lungs of other persons and is polluted by human bio effluents and other contaminants". Besides the above mentioned disadvantage, a total volume ventilation system is not offering individual control on the system, and the possibility of energy saving. Personalized ventilation concept is applicable for places where people spend most of their time sitting on a desk. The goal of this I. Cruceanu, is with the Civil Engineering, UTCB, Bd. Lacul Tei nr. 122 -124, Romania (Corresponding author: iunia.gligor@yahoo.com) I. Colda is with the Technical University of Civil Engineering Bucharest, Blvd Pache Protopopescu 66, Romania C. Maalouf, M. Lachi are with Laboratoire Thermomécanique-GRESPI, Faculté des Sciences, University of Reims Champagne Ardenne, BP1039, 51687, Reims, France ( Corresponding author: chadi.maalouf@univreims.fr). type of ventilation is to bring fresh air to the user and in doing so, to improve the thermal comfort of the occupants [START_REF] Barna | Modelling of thermal comfort conditions in Buildings[END_REF][START_REF] Grigoriu | Environmental Air Conditioned Requirements New Approach[END_REF][START_REF]Ventilation for acceptable indoor air quality[END_REF]. It also can give the user the possibility to adjust the direction and air flow rate. The system uses supply grilles located close to the breathing area of the user [START_REF] Ak | Personalized ventilation[END_REF], so that the occupant inhales the air from the unpolluted core of the supply jet (Fig 1). This means that the quality of the inhaled air is increased [START_REF] Ak Melikov | Impact of airflow interaction on inhaled air quality and transport of contaminants in rooms with personalized and total volume ventilation, Proceedinds of Healthy Buildings[END_REF], the risk of transmission of the infectious agents diminishes [START_REF] Bolashiikov | Methods for air cleaning and protection of building occupants from airborne pathogens[END_REF][START_REF] He | CFD Study of exhaled droplets tranmision between occupants under different ventilation strategies in a typical office room[END_REF][START_REF] Gao | Distribution on respiratory droplets in enclosed environments under different air distribution methods[END_REF] and the efficiency increases. Niu [START_REF] Niu | Experimental study on a chair-based personalized ventilation system[END_REF] found that over 80% of the inhaled air could be composed of fresh personalized air. The increase of the ventilation efficiency implies reduced outdoor supply air rates and reduced energy consumption [START_REF] Faulkner | Ventilation efficiencies and thermal comfort results of a desk-edge-mounted task ventilation system[END_REF]. There are two more strategies used to reduce energy consumption [START_REF] Seem | The impact of personal environmental control on building energy use[END_REF][START_REF] Bauman | Localized comfort control with a desktop task/ambient conditioning system: laboratory and field measurements[END_REF][START_REF] Schiavon | Energy-saving strategies with personalized ventilation in cold climates[END_REF][START_REF] Bauman | Localized comfort control with a desktop task/ambient conditioning system: laboratory and field measurements[END_REF][START_REF] Sekhar | Energy efficiency potential personalized ventilation system in thetropics[END_REF]: supplying fresh air only when the user is on his desk and the increase of the interior air temperature due to the fact that the personalized ventilation concept is based on the creation of a microclimate. Studies have shown that a personalized Fig. 1: Principle of desk mounted personalized ventilation system.

ventilation system can offer comfortable conditions even if the indoor temperature in the room is higher than the acceptable temperature. This is due to the fact that the most sensitive part on heat is the head. Sometimes with the classical ventilation systems due to the stratification, in the head zone the temperature could rise and become outside of the comfort zone. In alternative the PV system offers sufficient air movement around the head and the face to give the occupant a comfort sensation [START_REF] Gao | CFD Study on micro-environment around human body and personalized ventilation[END_REF][START_REF] Zhang | Modeling thermal comfort in stratified environments[END_REF][START_REF] Kwd | Local thermal sensation and comfort study in a field environment chamber served by displacement ventilation system in the tropics[END_REF][START_REF] Yang | The impact of personalized ventilation system on indoor air quality at different levels of room air temperature[END_REF][START_REF] Kaczmarczyc | Human response to personalized ventilation and mixing ventilation[END_REF].

The purpose of this study is to analyse the energy consumption of a personalized ventilation system and compare it to a traditional ventilation system in the case of a classroom situated in the city of Bucharest Romania, using the simulation environment SPARK [START_REF] Sowell | Efficient solution strategies for building energy, system simulation[END_REF][START_REF] Maalouf | Effect of moisture transfer on thermal inertia in simple layer walls. Case of a vegetal fiber material[END_REF][START_REF] Maalouf | Effect of moisture transfer on thermal inertia in simple layer walls. Case of a vegetal fiber material[END_REF][START_REF] Maalouf | Effect of Free Cooling on the Operation of a Desiccant Evaporative Cooling System[END_REF][START_REF] Tran Le | Transient hygrothermal behaviour of a hemp concrete building envelope[END_REF] suited to complex problems. We compared both cases for different supply air flow rates and temperatures. Energy performance and thermal comfort conditions are estimated using the heating and cooling energy consumption as well as the cumulated temperature frequency for each case. Finally, a parametric study is run in order to assess the effect of some problem physical parameters such as glazing solar heat gain coefficient, roof insulation thickness, room surface area, occupants number, internal and external heat transfer convection coefficients…

II METHODS

A Description of the classroom

The building is a classroom of 6.4x8.1 (51.84 m 2 ) and 3.5m height. It is located in Bucharest, Romania. Weather file is provided from the ASHRAE data base. The study period is for three monthes from April till June. The Western wall is an external wall and the southern façade is made of a double glazing with a solar heat gain coefficient of 40%. The other walls are considered as partitions and are in contact with conditioned spaces at 24°C.

The floor is composed of an 18 cm concrete slab, 10 cm polystyrene and 7 cm concrete. The ceiling is composed of 10 cm of polystyrene, 18 cm concrete and 2 cm plaster. The west wall is made of 18 cm bricks with an external insulation of 7 cm of polystyrene.

The partition walls are made of 15 cm of brick in sandwich between two layers of plaster, 2 cm each.

B. Occupancy and internal heat gains

We considered that the room is occupied by 31 persons from 8.00 to 18.00, five days a week. Each hour has a break of 8 minutes during which only 15 occupants remain in the class (monthes of April and May) and 8 persons in June. The infiltration rate depends on the internal pressure and is considered equal to 0.8 V/h when the ventilation system is off and 0.4 V/h when the system is on.

The lighting is on from 8h to 10h for the monthes of April and May (15 W/m 2 ).

C Description of the ventilation system

Two ventilation systems are compared: • A classical ventilation system for which the air flow rate is considered 7 l/s/occupant respecting the Romanian standard. This system is on from 8h till 19h in the evening apart from the week-end. • A personalized ventilation system for which the air is treated by an air-conditioning unit that regulates the supply air temperature and is adjusting the air flow according to the room's occupancy. There is no mechanical ventilation and the air is evacuated through room leakage. In both cases, room internal temperature varies freely without any regulation system.

D The simulation software and room model

Simulations are run within the Simulation Problem Analysis and Research Kernel (SPARK) [START_REF] Sowell | Efficient solution strategies for building energy, system simulation[END_REF][START_REF] Maalouf | Effect of moisture transfer on thermal inertia in simple layer walls. Case of a vegetal fiber material[END_REF][START_REF] Maalouf | Effect of moisture transfer on thermal inertia in simple layer walls. Case of a vegetal fiber material[END_REF][START_REF] Maalouf | Effect of Free Cooling on the Operation of a Desiccant Evaporative Cooling System[END_REF][START_REF] Tran Le | Transient hygrothermal behaviour of a hemp concrete building envelope[END_REF], an equation based modeling environment developed by the Simulation Research Group at Lawrence Berkeley National Laboratory. The advantages of this environment are its modularity and its syntax which is suited to parametric studies. Room is modelled using a nodal method which considers the room as a perfectly mixed zone characterized by a pressure, a temperature and a moisture concentration. It involves equations for air and moisture mass balance, heat balance for ambiance and equations describing heat transfer through the walls, additional convection between inside wall surfaces and room balance. The energy balance equation for the air zone can be written as:

North South East West p i T I V c Φ - Φ + Φ - Φ = ∂ ∂ + τ ρ ) ( Source Top Bottom Φ + Φ - Φ + (1)
where I is room thermal inertia . This equation involves heat fluxes through the envelope (from walls and openings) and internal heat sources. Energy balance is also applied to walls and is solved using a finite difference method in which wall is discretized into 5 nodes [START_REF] Maalouf | A study of the use of thermal inertia in simple layer walls and its application to the use of a vegetal fiber material in buildings[END_REF]. External heat transfer convection coefficient is taken equal to 16 W/m 2 °C and internal heat transfer coefficient is equal to 5 W/m 2 °C.

Concerning radiation heat exchange between room walls it was computed using the mean radiant temperature method in which the radiative flow between a wall and all the other walls of the room is written as:

( ) m r LW rad T T S h - = Φ int , ( 2
)
The value of h r is expressed by: Fig3: Variation of the energy consumption needed to cool the inlet air as a function of air flow rate and maximum supply temperature for both the personalized ventilation case (right) and the classical ventilation case (left).

E. Simulated cases

We compared the two ventilation systems from both energy consumption and thermal comfort sides. For the conventional ventilation system (VC), the inlet air flow rate was calculated according to the Romanian Standard. Two values are considered of 840 m 3 /h and 995 m 3 /h. The supply temperature may vary in the range of 18°C (minimal) and a maximal value that could be 18, 20, 22 and 24°C. We have analyzed a total of 8 cases.

For the personalized ventilation system (VP) several supply air flow rates are simulated -2,5 l/s; 4 l/s; 5,5l/s; 6,5 l/s; 7 l/s; 8 l/s and 10 l/s for occupant. Due to the fact that the system supplies the air to the breathing zone of the occupant, some restrictions must be imposed regarding the supply air temperature. The temperature will vary between a minimum of 20°C and an imposed maximum of 22, 24 or 26°C (the fresh air will be heated or chilled). We've studied a total of 21 cases.

For each case, 3 parameters were taken into consideration:

• The energy consumption HC needed to heat up the fresh air to reach its imposed minimum temperature. • The energy consumption CC needed to cool external air to its requested maximal temperature.

• The cumulated frequency that characterizes the interior comfort and is defined by the following relation :

FC= ( ) ( ) i t ref i T T t T δ ∑ - ) ( ( 5
)
T ref is the reference temperature taken to be 28°C for the classic ventilation and 30°C for the personalized ventilation.

( ) i T δ is the symbol of Kronecker and is equal to 1 if T i is superior to T ref , otherwise it is equal to 0. This frequency accounts for the degree-hours of thermal discomfort. The smaller it is, the greater thermal comfort is expected. The simulations are carried for April, May and June however, in order to initialize the calculations and to avoid the influence of the initial conditions, only the results of May and June are taken into consideration.

III RESULTS

Figure 3 shows the cooling energy consumption (CC) for both cases, personalized and classic ventilation, as a function of air flow rate and for different maximal air supply temperatures. As we can see in both cases, energy consumption increases with air flow rate and decreases with the growth of maximal air supply temperature. For the classic ventilation system and an air flow of 25 m 3 /h/person, the energy consumption varies from 182.9 kWh to 755.8 kWh when the maximum supply temperature is lowered from 24 to 18°C. Increasing the air flow from 25 to 30 m 3 /h/person brings an 18% increase in the energy consumption. For the case of the personalized ventilation and the same airflow rates, the consumption varies between 80.3 kWh and 262.7 kWh for a maximum temperature of 26 to 22°C. Due to the fact that the air, in this case, is introduced near the user, an inlet temperature of 18°C cannot be used. With a growth of 2°C, cooling energy consumption decreases about 50%.

Figure 4, presents the heating energy consumption variation (HC) needed for both the classical and personalized ventilation systems to heat up the fresh air to a minimum supply temperature of 18°C (for the CV) and 20°C (for the PV). We notice that for both cases this consumption is not function of maximum supply temperature. It increases with supply airflow rate and it is higher for the personalized ventilation system. For 7 L/s/person it is 134 kWh for the (PV) whereas it is 89 kWh for the (CV). Figure 5 shows the variation of the cumulated frequency for both cases and for different air flow rates and maximum air supply temperatures. Higher air flow rates or lower supply air temperatures lead to a decrease of the cumulated frequency indicating that the internal temperature is lowered and the thermal comfort is improved.

For the classical ventilation case, the FC value is very high and it has satisfying values only for an inlet temperature of 18°C (both air flow rates) or a 20°C and an air flow of 30 m 3 /h/person. In the case of personalized ventilation the smaller values are for a maximal temperature of 22°C and for air flows larger than 6 l/s/person. For 24°C the interior temperature is higher than 28°C for the whole week and even reaches 32.2°C. The cumulated frequency is, in this case, 315 (° hours). When the inlet temperature is 18°C, the maximum interior temperature is 29.4°C and the frequency 27.5 (°hours). The energy consumption for cooling varies from 183 kWh to 755.8 kWh. To these values, 88.8 kWh (HC) must be added to take into account the energy needed for the air when its temperature is below 18°C.

When comparing energy consumption for the same air flow and the same maximal supply temperature for both systems, we notice that cooling energy is very higher for the classical case than the personalized ventilation case and that heating energy is lower for the classical case. The difference comes from the fact that in the personalized ventilation system the minimum supply air temperature is 20°C while for the traditional system it is 18°C. These results confirm Schiavon results [START_REF] Maalouf | A study of the use of thermal inertia in simple layer walls and its application to the use of a vegetal fiber material in buildings[END_REF][START_REF] Schiavon | Energy analysis of the personalized ventilation system in hot and humid climates[END_REF] that personalized ventilation is suited to hot climates and not to cold climates where heating energy consumption is higher than the classical ventilation system case. In our case, as the system is used for cooling, total energy consumption is reduced from 844.6 kWh to 400 kWh when using personalized ventilation system with a supply temperature of 22°C and an air flow rate of 7 l/s/person, with about 52% of reduction (FC=50°hrs). For a supply temperature of 24 °C and an air flow rate of 8 l/s/person, the energy consumption is about 327 kWh and the FC is about 64.5 °hrs which is also acceptable.

IV SENSITIVITY ANALYSIS

In this part we have decided to realize a sensitivity analysis to be able to see the influence of different parameters of the model on our results. The analysis will be made by comparison to the reference case (Fig 8) -PV air flow rate 8 l/s, maximum supply air temperature of 24°C, minimum air supply temperature 20°C (Fig 7) and an occupancy of 31 persons.

The first parameter that we decided to analyze is the number of occupants in the classroom.

A. Number of occupants in the classroom

Because we are talking about the ventilation system, the number of occupants present in the classroom will have an immediate impact on the energy consumption. We have varied the number of occupants of ± 20 % from the reference case number. The values of the results are presented in Table1.

As we can see if the number of people present in the classroom increases from 31 to 37 persons, the energy consumption will also rise about 18.36%. If the number decreases to 24, the consumption will decrease about 18%. It is also important to see the influence on the cumulated frequency which gives us an idea about the conditions inside the classroom. The value of the cumulated frequency diminishes when the number of occupants increases and has a small increase when the occupation number decreases. This is due to the fact that when occupancy increases, more fresh air is supplied to the room and its mean temperature slightly decreases, so the cumulated frequency will diminish about 2%. For the case with 25 persons we can observe that the increase for the cumulated frequency does not follow the same pattern as the cooling or heating consumption. The percent of increase is around 7.5%.

B .Solar heat gain coefficient

The second parameter we choose to vary is the solar heat gain coefficient (SHGC) for the south facing glazing. Again we selected the same interval of variation ±20 %. The values presented in Table 1 show that for the same airflow rate the SHGC has a great influence on the cumulated frequency and in consequence on the interior air temperature. An increase of the SHGC of 20% gives a value of the cumulated frequency of 12.2 (°hours) which represents a decrease of 81%. A decrease of 20% of SHGC gives us a value of FC of 175.84. This means that the cumulated frequency almost tripled its value (x2.72). So we can say that even if the SHGC does not affect the energy consumption, it has a direct impact on the internal thermal comfort conditions.

C.Roof insulation thickness

As we can see in the table below the insulation thickness of the roof is of little importance for the cumulated frequency. The variation is small compared to the reference case (4.89% for -20% insulation thickness and 6.99% for + 20% insulation thickness). As insulation thickness increases, internal heat is kept inside the room and evacuated slowly through the roof and thus the indoor temperature increases slightly

D.Exterior convection coefficient

Another parameter we decided to vary, is the value of the exterior convection coefficient. An increase of the exterior convection coefficient of 20% will slightly improve the interior comfort. Heat is evacuated faster to the outdoor ambiance and room temperature decreases. Cumulated frequency decreases about 4.9%. Decreasing the external convective heat transfer coefficient 20% of its value, leads to an increase of the interior temperature and a growth of 7% in the cumulated frequency value.

E. Interior convection coefficient

When varying the internal convection coefficient, the cumulated frequency follows the same pattern as for the external convection case. When the internal heat convection coefficient increases about 20%, the cumulated frequency diminishes. An increase of 20% of internal heat convection coefficient leads to a decrease of about 8.7% of the cumulated frequency because heat is evacuated fastly from the room. We notice that a decrease in the convection coefficient will slightly impact the cumulated frequency because indoor temperature won't increase any more. It only decreases slowly.

F. Outdoor air temperature

Table 1 shows the effect of variating outdoor temperature of ±10%.

We notice that an increase of 10% of the outdoor air temperature has a great impact on the energy consumption. For the heat energy consumption it is reduced about 32.3% compared to the reference case. For the cooling energy consumption it increases and the difference is more important. The cooling consumption has almost doubled. Total energy consumption (the sum of HC and CC) increases 40% compared to the reference case. In what concerns the cumulated frequency, its value will increase from 64.5 to 148 (°hours). Decreasing the outdoor temperature about 10% will raise the heating energy consumption more than 50%, and reduce the cooling energy consumption of about 60%. The total energy consumption is reduced about 10%. The cumulated frequency also decreases to 17 (°hours) meaning that internal comfort is improved.

G. Classroom dimensions

We were also interested to see the influence of the dimensions of the room on interior classroom temperature. For this case we decided to vary both the length and the width of the classroom due to the fact that one of the exterior walls is a glass wall. In this case the influence on the cumulated frequency will be greater, as we can see in the Table 2 above.

We can see that an increase of the window surface of 10 % raises the value of the cumulated frequency about 19 %, and increasing the width of 10% will increase cumulated frequency of almost 11% because no more solar energy will enter as in the case of the length.

V. CONCLUSIONS AND PERPECTIVES

In this paper, energy consumption of two ventilation systems was analyzed and compared: classical ventilation system and a personalized ventilation system. The system is used to cool a classroom in Bucharest, Romania. Both systems were modelled using the simulation environment SPARK. Simulations are used to evaluate energy consumption and thermal comfort conditions. Our results suggest that when compared to classical ventilation system, personalized ventilation can provide comfortable conditions and reduce energy consumption of about 60%.

A sensitivity analysis was made in order to see the influence of different physical parameters on the energy consumption and on the interior air temperature. As we could see, the variation of the occupants number had an influence on the energy consumption but it did not had a major influence on the cumulated frequency -consequently on the interior air temperature.

The parameter that has a great influence on the interior temperature is the solar heat gain coefficient Energy consumption is mainly affected by outdoor temperature variations. Simulations have shown that when temperature increases (weather becomes hotter) energy consumption will increase drastically and also thermal conditions will get worse.

These results should be taken carefully as more analysis is to be done for the personalized system in order to analyze air flow around occupants.

Fig.. 2

 2 presents the sensible heat load of the room due to lighting and occupants (monthes of April and May).

Fig 2 :

 2 Fig 2: Heat load in the classroom due to light and occupancy for the monthes of April and May.

  σ 0= 5,67*10 -8 W m -2 K -4 -Stephan Boltzmann Constant Where T m is the mean radiant temperature of the walls and is given by:

Fig 4 :

 4 Fig 4: The energy consumption needed to heat the inlet air for both PV (left) and CV (right) cases.

Fig 5 :

 5 Fig 5: The variation of the cumulated frequency (FC) for both cases PV (top) and CV (bottom) cases.

Fig 6 :

 6 Fig 6: The variation of the interior temperature for different air flows in classical ventilation case during the month of june for the hottest week of the month

Fig 7 .

 7 Fig 7. PV air temperature as a function of the outdoor temperature (case when T supply is between 18°C and 24°C

TABLE 1 -

 1 THE IMPACT OF THE VARIATION OF THE OCCUPANTS NUMBER, SOLAR HEAT GAIN COEFFICIENT AND ROOF'S INSULATION THICKNESS RELATIVE TO THE REFERENCE CASE

		Heating capacity	Cooling capacity	Cumulated frequency
	Parameters
		(kWh)	(kWh)	(° Hours)
	Reference case 24°C, 8l/s	153.1	174	64.5
	Ocupants +20%	181.1	205.95	63.193
	Occupants -20%	125	141.88	69.33
	SHGC +20%	153.1	174	175.84
	SHGC -20%	153.1	174	12.2
	Insulation thickness +20%	153.1	174	67.5
	Insulation thickness -20%	153.1	174	60.6
	Ext. conv. coefficient +	153.1	174	61.34
	20%
	Ext. conv. coefficient -	153.1	174	69
	20%
	Interior convection	153.1	174	58.9
	coefficient +20%
	Interior convection	153.1	174	64.9
	coefficient -20%
	Outdoor air temperature	103.5	356	148
	+10%
	Outdoor air temperature	230.46	64.8	16.6
	-10%

TABLE 2 -

 2 THE IMPACT OF THE VARIATION OF THE DIMENSIONS OF THE ROOM RELATIVE TO THE REFERENCE CASE

		Heating capacity	Cooling capacity	Cumulated frequency
	Parameters
		(kWh)	(kWh)	(° Hours)
	Reference case 24°C, 8l/s	153.1	174	64.5
	Width +10%	153.1	174	71.52
	Width -10%	153.1	174	52.23
	Length +10%	153.1	174	76.95
	Length -10%	153.1	174	48.75

Issue 2, Volume 7, 2013