Abstract
For $f$ an entire transcendental map with a univalent Baker domain $U$ of hyperbolic type I, we study pinching deformations in $U$, the support of this deformation being certain laminations in the grand orbit of $U$. We show that pinching along a lamination that contains the geodesic $λ_{\infty}$ (See Section 3.1) does not converges. However, pinching at a lamination that does not contains such $λ_{\infty}$, converges and converges to a unique map $F$ if: the Julia set of $f$, $J(f)$ is connecte
Results & Lemmas (10)
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Theorem 2.
Theorem 2. Let f be an entire transcendental map satisfying: (a) f has a univalent Baker domain of hyperbolic type I, 1
Theorem 2. Let f be an entire transcendental map satisfying: (a) f has a univalent Baker domain of hyperbolic type I, 1
Lemma 1.
Lemma 1. If Λ does not contains λ∞and γ1, γ2 are in R(Λ), then ¯γ1 T ¯γ2 = ∅
Lemma 1. If Λ does not contains λ∞and γ1, γ2 are in R(Λ), then ¯γ1 T ¯γ2 = ∅
Lemma 2.
Lemma 2. Assume that ht ⇒H, then for any γ ∈R(Λ), limt→1 (diamsht(¯γ)) = 0
Lemma 2. Assume that ht ⇒H, then for any γ ∈R(Λ), limt→1 (diamsht(¯γ)) = 0
Lemma 3.
Lemma 3. If f satisfies conditions (a) and (b) above, the diameter of any sequence of elements in V(Λ) tends to 0 if λ∞/∈Λ.
Lemma 3. If f satisfies conditions (a) and (b) above, the diameter of any sequence of elements in V(Λ) tends to 0 if λ∞/∈Λ.
Lemma 4.
Lemma 4. Let f be a entire transcendental map with a univalent Baker domain U of hyperbolic type I. Then, the pinching process along the…
Lemma 4. Let f be a entire transcendental map with a univalent Baker domain U of hyperbolic type I. Then, the pinching process along the lamination λ∞does not converges.
Theorem 1.
Theorem 1. Let f be a entire transcendental map with a periodic univalent Baker domain U of hyperbolic type I. If Λ is a Baker lamination…
Theorem 1. Let f be a entire transcendental map with a periodic univalent Baker domain U of hyperbolic type I. If Λ is a Baker lamination on U that contains λ∞, then, the pinching process along the lamination Λ does not converges. 6. pinching along general laminations Let us denote by area(A) the plane area of a set A.
Lemma 5.
Lemma 5. area(R(Λ)) = 0
Lemma 5. area(R(Λ)) = 0
Theorem 2.
Theorem 2. Let f be an entire transcendental map satisfying the following prop- erties: (a) f has a univalent Baker periodic domain U of…
Theorem 2. Let f be an entire transcendental map satisfying the following prop- erties: (a) f has a univalent Baker periodic domain U of hyperbolic type I, (b) the postcritical set P(f) is a positive distance away from the Julia set. (c) J(f) is thin at ∞. Then for Λ any Baker lamination in U which does not contains λ∞, the pinching process along R(Λ) converges uniformly to an entire transcendental map F which exhibits in its Fatou set a family of bounded simply connected wandering domains, disj
Theorem 3.
Theorem 3. If Λ is as in (A), then U becomes in the limit of the deformation a Baker domain with a family of wandering domains attached to…
Theorem 3. If Λ is as in (A), then U becomes in the limit of the deformation a Baker domain with a family of wandering domains attached to its boundary. If Λ is as in (B), then, only a family of wandering domains appears. Figure 2. A Baker domain U with a Baker lamination (case A) and the result of the pinching: a new Baker domain and four dif- ferent wandering domains. 7. Teichm¨uller space and Hurwitz class For the set of quasiconformal homeomorphisms of C, define an equivalence rela- tion ∼by
Proposition 1.
Proposition 1. Assume that there is a sequence (ft) ∈T eich(U/f) such that (ft) ⇒F then, if there are new Fatou periodic domains of F, they…
Proposition 1. Assume that there is a sequence (ft) ∈T eich(U/f) such that (ft) ⇒F then, if there are new Fatou periodic domains of F, they are univalent Baker periodic domains.