Invariance of domain H η,ρ under g λ,µ

which also yields the result. In particular, the function hδ,µ(x) equals zero at x= 2 ln

δ (1−µ)|λ|

.

Proposition 2.29. For every λ∈C^∗, δ∈(0,1) andx < σµ, hδ,µ(x) is monoton-ically decreasing.

Proof. For simplicity in this proof we use the notation A:= _(1−µ)|λ|^δ . Notice that A is always positive. Then rewritingh_δ,µ(x) =√

A²e^−2x−x² we have h⁰_δ,µ(x) = −2A²e^−2x−2x

2hδ,µ(x) = −A²e^−2x−x

√

A²e^−2x−x².

To show thath_δ,µ(x) is monotonically decreasing, we show−A²e^−2x−xis negative.

Since x ≤σµ <0, we writes=−x > 0 and distinguished between the following two cases.

Case 1: x <−e <2 lnA. Thens > eand we have 0<3−e= 3 lne−e <3 lns−e as well as lns <4 lns+ 2 lnA <2s+ 2 lnA. The latter inequality is equivalent to s < A²e^2s which implies in turn −A²e^−2x−x <0.

Case 2: x <2 lnA <−e. Then we obtain for positive values−s <2 lnA, implying that lns < s = 2s−s <2s+ 2 lnA. Again, the latter inequality is equivalent to s < A²e^2s providing the same result as in the first case.

We remark that the boundary of M_δ,µ is symmetric with respect to the real axis, as the domainM_c,µdoes. Therefore∂M_δ,µis monotonically decreasing for values above the real axis and monotonically increasing for values above the real axis.

The previous proposition implies in particular for values of x≤σ_µ that ∂M_δ,µ is an unbounded simple curve.

2.4.2 Invariance of domain H_η,ρ under g_λ,µ

We devote this subsection to invariance of the smaller domain H_η,ρ := H_η ∪ M_ρ contained in M_δ,µ. It can be the case that M_δ,µ is mapped completely into H_η,ρ, after several iterations, or even that M_δ,µ intersects the Julia set, in which

2.4. Absorbing domainH_η,ρ 65

case is the reasons not to be invariant. The occurrence of one behaviour or the other depends on the given parameter λ and its relations with µ and δ, as we show subsequently. In other words, there may exist z∈ M_δ,µ\ H_η,ρ belonging to the Julia set. Therefore we restrict to the smaller region M_ρ and construct an unbounded domain H_η contained in a half plane located at the left side of M_ρ where we can guarantee the invariance under the functiong_λ,µ.

LetM_δ,µbe the domain defined in the previous subsection by the expression (2.18) and bounded by the curves±h_δ,µ(x) defined in Equation (2.20). Consider a fixed parameterδ = ¹₂ and write the discriminant of h1/2,µ as the restriction ofM_δ,µto values ofxin the interval

straight lines that join the origin with the pointszρ,± =

1− _2µ¹ ,±q

ρµ(1−_2µ¹ )

. Notice that the pointszρ,± lie on the boundary ofM_ρ. We define the domain as

H_η := for all µ ≤ µ1 < ¹₂. Under this assumption the existence of the attracting fixed point z_µ is guaranteed. However, we show below in Lemma 2.32 that we require a smaller boundary value of µ such that the invariance of the absorbing domain is assured for any given λ. In order to state the theorem on invariance we give here the necessary conditions on the parameters and remark that for these µ’s

66 2. Dynamical description of the familyg_λ,µ

Figure 2.9: Invariant absorbing domainH_η,ρ defined as the union of two regions: firstlyH_η, which lies between the linesη_µand−η_µ, and secondly M_ρ, which is a restriction of M_1/2,µ for Rez >1−_2µ¹ .

the existence of the attracting fixed point is automatically fulfilled. Let µ2 be sufficiently small that

|λ|< e

1 2

1 µ−2

µ²

1−µ for allµ≤µ₂. (2.24) Now we obtain for the domains and the parameters as defined above the following result.

Theorem 2.30 (Invariance of absorbing domain). Let H_η,ρ := H_η ∪ M_ρ. Then the domainH_η,ρ is invariant undergλ,µfor everyλ∈C^∗ and for everyµ∈(0, µ2], with µ2 =µ2(λ).

We first present some lemmas needed for the proof. The proof of Theorem 2.30 follows in conclusion. A first tool to prove the invariance of the absorbing domain H_η,ρ is achieved by showing that two subdomains of H_η,ρ are mapped into H_η,ρ. The first region is given by the intersection M_δ,µ with a right half-plane and we prove that it is mapped to the left. The second region is the intersection of H_η,ρ with an upper and a lower half-planes which are mapped towards the real axis.

2.4. Absorbing domainH_η,ρ 67

Letδ ∈(0,1). We define H_Re:=M_δ,µ∩

z∈C: Rez >1− 1 µ + δ

µ

and H_Im:=M_δ,µ∩

z∈C:|Imz|> δ µ

.

Lemma 2.31. For the domains defined above we obtain

• If z∈ H_Re, then Reg_λ,µ(z)<Rez.

• If z∈ H_Im, then |Img_λ,µ(z)|<|Imz|.

Proof. Firstly consider z∈ H_Re. In particularz is also contained inM_δ,µ, there-fore (1−µ)|λze^z| ≤δ is satisfied. On the one hand we have z∈ H_Re if and only if−µRez≤1−µ−δ which is equivalent to

(1−µ)(Rez−1) +δ≤Rez.

On the other hand, Regλ,µ(z) ≤ (1−µ)Re (z−1) + (1−µ)|λze^z| holds. Since z∈ M_δ,µ this implies

Reg_λ,µ(z)≤(1−µ)(Rez−1) +δ.

In the first case equality holds for z with Rez = 1− ¹_µ +_µ^δ, that is, on the left side of H_Re. In the second case we obtain equality only for z ∈R∩∂M_δ,µ, that is, on the right side of H_Re. We can resume that both equalities never occur simultaneously forz∈C. This assures the strict inequality Reg_λ,µ(z)<Rez.

Secondly letz∈ H_Im. We prove the result for Imz≥ ^δ_µ and it analogously follows for Imz≤ −_µ^δ. The inequality Imz ≥ _µ^δ is equivalent to (1−µ)Imz+δ ≤Imz.

Then such a point also satisfies Img_λ,µ(z) ≤ (1−µ)Imz+ (1−µ)|λze^z|. This yields that

Imgλ,µ(z)≤(1−µ)Imz+δ.

In this case equality holds if the intersection of ∂H_Im with ∂M_δ,µ lies on iR. However it is not possible, because Rez <0 for allz ∈∂M_δ,µ. Hence we obtain a strict inequality as well.

In order to proof that the domainH_η,ρ is invariant with respect tog_λ,µ, we firstly show that gλ,µ(∂Hη,ρ) ⊂ Hη,ρ. The latter statement follows almost immediately from the lemma above if H_Re∩ H_Im 6= ∅. A non-empty intersection is assured

68 2. Dynamical description of the familyg_λ,µ

Figure 2.10: Domain M_δ,µ together with regions H_Im and H_Re, where

|Img_λ,µ(z)|<|Imz|and Reg_λ,µ(z)<Rez respectively. The intersection of H_Reand H_Im is non empty forδ ≤ ¹₂.

whenh_δ,µ

1−¹_µ+_µ^δ

> _µ^δ. Forδ= 1 the inequality reduces toh_δ,µ(1)> _µ¹ which is fulfilled if and only if e⁻² >|λ|²(1−µ)²(1 + _µ¹2). This is a strong restriction, hardly satisfied by an arbitrary parameter λ. In other words, we can always find values ofλsuch that the intersection is empty, even when we take the limitµ→0.

In contrast to this, consider now a small value of δ ∈ (0,1), for example δ = µ.

Then the intersection of H_Re and H_Im is non empty, since it is equivalent to 2q

1

µ−1 >1. But with this, the disc D((1−µ)(z−1), δ) containing g_λ,µ(z) as defined for points z∈ M_δ,µcollapses to a point as we take the limit δ =µ→0.

Following these arguments, we may restrict the value ofδ so that we can continue with the analysis for arbitrary values of λ∈C^∗ and µ∈(0, µ2]. Letδ ≤ ¹₂ and µ2

be sufficiently small that Equation (2.24) is fulfilled for allµ≤µ₂. Then we have as well that _|λ|¹2 > e

2−_µ¹

(1−µ)² µ⁴ .

We claim that H_Re is not empty for δ = ¹₂. We prove it by showing that ρµ

1−_µ¹ + ^δ_µ

> 0 for values of µ2 and λ as above. After some computation,

2.4. Absorbing domainH_η,ρ 69

In what follows, we consider the fixed value δ = ¹₂ and denote the function h_δ,µ defined in Equation (2.20) simply byhµ. Recall thathµ(x) =p

On the other hand, the latter expression is larger than _2µ¹ if and only if 1

An important consequence of choosing the parameterδ= ¹₂ is stated as follows.

Corollary 2.33. For δ= ¹₂ the domainsH_Re and M_ρ coincide.

70 2. Dynamical description of the familyg_λ,µ

A second tool in the proof of the invariance of H_η,ρ is to characterise points z ∈ M_δ,µ that are at a distance ¹₂ from the point (1−µ)(z−1). We prove in the following lemma that a point z belongs to such a disc if and only if z lies in a neighbourhood of the attracting fixed pointz_µtangent toH_Re andH_Im as it is shown in Figure 2.11. Recall that forδ = 1/2 these domains simplify by

H_Re⊂

Proof. Letz∈ M_1/2,µ. Using definition the definition ofM_1/2,µ we obtain thatz lie in the disc D µ≤µ2, as stated in Theorem 2.30. Furthermore, we show the results for pointsz with Imz >0 and they analogously apply forzwith Imz <0, following the same argument for the functions−η_µ(x) and −h_µ(x).

Proof of Theorem 2.30. Now we prove the invariance of the domain H_η,ρ. It is clear that the attracting fixed point z_µ is contained in H_η,ρ, since z_µ lies in the disc D

Letz_bbe an arbitrary point in the boundary ofH_η,ρand denote the disc containing its image gλ,µ(zb) byDb := D (1−µ)(zb−1),¹₂

2.4. Absorbing domainH_η,ρ 71

Figure 2.11: Invariant fundamental domain H_η,ρ defined for δ = 1/2, µ≤µ2. The discD

1− ¹_µ,_2µ¹

is tangent to the boundaries of the domains H_Reand H_Im.

∂M_ρ is monotonically decreasing (reps. monotonically increasing) for point with imaginary part greater (resp. lower) than zero as proved in Proposition 2.29, we have that g_λ,µ(z_b) ∈ M_ρ. In particular this implies that the whole disc D_b is as well contained inM_ρ.

Secondly, writez_ρ:=

1−_2µ¹ ,q

ρ_µ(1−_2µ¹ )

for the boundary point lying on∂M_ρ such that Rezρ < Rez for every z ∈ M_ρ as it is marked in Figure 2.11. Using Lemma 2.32z_ρ∈ H_Re∩ H_Im. Hence Reg_λ,µ(z_ρ)<Rez_ρ and Img_λ,µ(z_ρ) <Imz_ρ which implies that g_λ,µ(z_ρ)∈ H_η,ρ.

Thirdly, consider a point zb ∈ ∂H_η. Then it satisfies zb = xb+iηµ(xb) and we refer the reader to Figure 2.12 for a sketch of the following computations. Recall that every z_b ∈∂H_η ⊂ M_1/2,µ satisfies (1−µ)|λz_be^zb|< ¹₂. Hence we obtain

Regλ,µ(zb) = (1−µ)(xb−1) + (1−µ)Re (λzbe^zb)

≤(1−µ)(x_b−1) + (1−µ)|λz_be^zb|<(1−µ)(x_b−1) +1 2. Analogously Imgλ,µ(zb) <(1−µ)Imzb +¹₂ holds. We prove that gλ,µ(zb) lies in H_η by showing that Img_λ,µ(z_b) < η_µ(Reg_λ,µ(z_b)). Since z_b ∈ ∂H_η, it suffices to

72 2. Dynamical description of the familyg_λ,µ Recall that the parameter λsatisfies _|λ|¹2 > e

2−_µ¹

(1−µ)²

µ⁴ . Then the right side of the latter inequality is greater than

µ

Finally comparing the latter two computations, the Inequality (2.25) is satisfied provided 1 + (2µ−1)² < _µ¹2 which is valid for everyµ < ¹₂.

Fourthly, we show for completeness that the point z_RI = 1− _2µ¹ +i_2µ¹ , lying at the left bottom ofH_Re∩ H_Im as marked in Figure 2.11, is also mapped inside. In particular, it holds

The arguments presented during the whole proof are analogously satisfied for points of H_η,ρ with negative imaginary part. Due to continuity, the intersection of the real axis with H_η,ρ is also mapped inside of H_η,ρ by g_λ,µ, completing the proof

We have shown in Theorem 2.30, that the domainH_η,ρ is forward invariant. Now we prove that it is an absorbing domain for A_µ. For this it is left to show that given any compact setK in the basin of attractionA_µ, there exists ann∈Nsuch thatg_λ,µⁿ (K)⊂ H_η,ρ. We show it in the following result.

2.4. Absorbing domainH_η,ρ 73

Figure 2.12: For a boundary pointz_b=x_b+iηµ(x_b) on∂H_η the disc containing its imageg_λ,µ(z_b) denoted byD_b :=D (1−µ)(z_b−1),¹₂ is mapped inside of H_η,ρ.

Proposition 2.35. The domain H_η,ρ⊂ A_µ is absorbing for g_λ,µ.

Proof. LetK be any compact set contained in the basin of attractionA_µ. LetDµ

denote the discD

(1−_µ¹),_2µ¹

⊂ A_µand recall thatDµ⊂ H_η,ρis tangent toH_Re and H_Im. In particular D_µ contains the attracting fixed point z_µ and all points z inD_µ convergence to z_µ under iterates. Even more, for every z ∈ A_µ exists an n ∈ N such that gⁿ_λ,µ(z) ∈ Dµ. Hence, A_µ ⊂ S

n∈Ng⁻ⁿ_λ,µ(Dµ). In particular, this covers the compact set K. Then there exists a finite subcoveringS

m∈Ig_λ,µ^−m(Dµ) ofK with a finite index setI ⊂ {0, . . . , n₀}. This implies thatgⁿ_λ,µ⁰ (K)⊂D_µ and in particular it is also contained inH_η,ρ for somen≤n0.

It was proved by Lauber in [48] thatM_δ contains all critical values, with at most one exception. In comparison to this and together with the properties of critical points and values presented in the previous section we show that a finite number of the critical values of g_λ,µ are also contained inH_η,ρ. More precisely we obtain the following result.

Corollary 2.36. A finite number of critical values vk are contained in the ab-sorbing domainH_η,ρ while all critical points c_k lie outside of it.

Proof. As we showed at the beginning of this section in Equation (2.19), the boundary of M1

2,µ is approximately a translation by one to the left of ∂M_c,µ where the critical points are located, yielding the second statement. Since v_k ∈

74 2. Dynamical description of the familyg_λ,µ

D c_k−2,¹₂

using Lemma 2.25, then that all critical values are in M1

2,µ by def-inition of the domain. In other words, the critical values are in the preliminary domain M1

2,µ, before we take the restriction to the smaller domain M_ρ. With this, we obtain only a finite number of them in H_η,ρ, that is only the first |k|

contained inM_ρ.

In other words, Corollary 2.36 implies that g_λ,µ is univalent in the absorbing domain. However, since the Baker domain is not univalent, it follows that its boundary is not a Jordan curve, using a result due to Baker and Weinreich men-tioned in the introduction. In the following sections we use the location of the critical points and values as stated in the corollary above to construct further structure in the basin of attraction outside the absorbing domain. This enable us a better understanding of the Julia set ofg_λ,µ.

Im Dokument Approximation of Baker domains and convergence of Julia sets. (Seite 70-80)