Univalent Baker domains and boundary of deformations

Theorem 2. Let f be an entire transcendental map satisfying: (a) f has a univalent Baker domain of hyperbolic type I, - (b) the postcritical set, is a positive distance away from the Julia set and - (c) J(f) is thin at $\infty$ . Then for $\Lambda$ any Baker lamination in U which does not contains $\lambda_{\infty}$ , the pinching process along $\mathcal{R}(\Lambda)$ converges uniformly to an entire transcendental map F which exhibits in its Fatou set a family of bounded simply connected wandering domains, disjoint from the postcritical set. The hypothesis of the theorem allow the map f to be expanding in the Julia set according to a theorem of Stallard ([St], Theorem A) and so it is weakly hyperbolic (Section 2.3), hence the convergence techniques in the paper of Haissisky and Tan Lei [HT] can be used in this case. This is the main reason to have chosen such kind of Baker domains. In section 7, we study pinching deformations as seeing in the marked Hurwitz space of the map f, here denoted by $M_f$ and which is defined as the space of all marked maps topologically equivalent to f (Definition 4). We include the Teichmüller space of f, Teich(f) into $M_f$ . Theorem 1 shows that pinching along $\lambda_{\infty}$ does not converges to a point in $M_f$ , hence Teich(f) is not compactly contained in $M_f$ . Pinching along the other Baker laminations defines (by Theorem 2) points at the boundary of Teich(f) contained in $M_f$ . Finally, it is shown that if we take a sequence of deformations in the slice Teich(U/f) of Teich(f), with the property that it converges uniformly to an entire transcendental map at the boundary of Teich(f) (as subset of $M_f$ ) then, the only development of U towards a periodic Fatou domain (in case it exists) is to a new periodic univalent Baker domain.

Lemma 1. If $\Lambda$ does not contains $\lambda_{\infty}$ and $\gamma_1$ , $\gamma_2$ are in $\mathcal{R}(\Lambda)$ , then $\bar{\gamma_1} \cap \bar{\gamma_2} = \emptyset$

Lemma 2. Assume that $h_t \rightrightarrows H$ , then for any $\gamma \in \mathcal{R}(\Lambda)$ , $\lim_{t\to 1} (diam_s h_t(\bar{\gamma})) = 0$ Proof. Notice that since the ellipse field is parallel to the geodesic $\gamma$ , then $h_t(\gamma) \in W_t$ , is a geodesic. Divide $\gamma$ in segments $s_i$ of the same length, then $l_{W_t}(s_i) < l_{W_r}(s_i)$ if t < r. Since $h_t$ converges, $l_{W_t}(s_i)$ converges to zero as $t \to 1$ , otherwise we could construct an ellipse field over the geodesic and we can apply a pinching process again. Hence, the two points of $End(\gamma)$ must collide and we conclude that $lim_{t\to 1}$ $(diam_sh_t(\bar{\gamma})) = 0$ .

Lemma 3. If f satisfies conditions (a) and (b) above, the diameter of any sequence of elements in $V(\Lambda)$ tends to 0 if $\lambda_{\infty} \notin \Lambda$ .

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 $\lambda_{\infty}$ does not converges. Proof. Let us assume that $f_t \rightharpoonup F$ , by Lemma 2, $\lim_{t \to 1} (\dim_s h_t(\overline{\lambda_\infty})) = 0$ which implies that $h_t(\overline{\lambda_\infty}) \to \infty$ . For $P \in \partial U$ the repulsive fixed point of f, let $P_t = h_t(P)$ . Then $P_t \in \partial(h_t(U))$ is a boundary point of $\lambda_\infty$ other than $\infty$ (along the process of pinching) and we have that $P_t \to \infty$ , as $t \to 1$ . Since $P_t \in J(f_t)$ , by the f-invariance of the pinching, we have that any point in the set $f_t^{-n}(P_t)$ tends to $\infty$ under $h_t$ , for any $n \geq 0$ and any t < 1. However, the set $\bigcup_n f_t^{-n}(P_t)$ is dense in $J(f_t)$ , then by uniform convergence we have that $J(F) = \infty$ . That is a contradiction. As a direct consequence of this lemma, we have the following theorem.

Theorem 1. Let f be a entire transcendental map with a periodic univalent Baker domain U of hyperbolic type I. If $\Lambda$ is a Baker lamination on U that contains $\lambda_{\infty}$ , then, the pinching process along the lamination $\Lambda$ does not converges.

Lemma 5. $area(\mathcal{R}(\Lambda)) = 0$

Theorem 2. Let f be an entire transcendental map satisfying the following properties: - (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 $\infty$ . Then for $\Lambda$ any Baker lamination in U which does not contains $\lambda_{\infty}$ , the pinching process along $\mathcal{R}(\Lambda)$ converges uniformly to an entire transcendental map F which exhibits in its Fatou set a family of bounded simply connected wandering domains, disjoint from the postreritial set.

Theorem 3. If $\Lambda$ 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 $\Lambda$ 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 different wandering domains.

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. Proof. Since F ∈ M^f then F is entire transcendental. If F has new Fatou periodic domains, then by the uniform convergence of (ft) to F, it has to be a periodic univalent domain without singularities, hence a univalent periodic Baker domain. The singular set of F is contained in the persistent components of the Fatou domain. Acknowledgments: To Peter Makienko who kindly explain us his early work on pinching, to Gwyneth Stallard for bringing our attention to her paper. To Memox for the figures. To PAPIIT grant IN105306.

Definition 1. (Baker domain) For f an entire transcendental map, U a periodic component of the Fatou set of period p, is called a Baker domain if for all $z \in U$ , $f^{np}(z) \to \infty$ as $n \to \infty$ . It was proved by Baker in [Ba] that all Baker domains for entire transcendental maps are simply connected. In [EL] it is established that if $Sing(f^{-1})$ is bounded then f has not Baker domains. It is not necessarily for a Baker domain to have any of the singular values inside. A Univalent Baker domain is a periodic Baker domains on which $f^p$ is univalent and these are the kind of domains that we are interested in this paper. For entire transcendental maps, a classification of this domains is given by Baransky, Hendriksen and Fagella [BF], [FH1] as follows. Assume that f has an invariant Baker domain U, then there exist a point $\xi \in \hat{\mathbb{C}}$ such that the backward iterates under $(f|_U)^{-1}$ of all points in U tend to $\xi$ through the same access (they call it the backward dynamical access), and one of the following cases occurs: - (a) If $\xi \neq \infty$ is a fixed point in the boundary of U, then U is of hyperbolic type I. - (b) If $\xi = \infty$ but the backward dynamical access is different than the forward dynamical access, then U is of hyperbolic type II. - (c) If $\xi = \infty$ but the backward dynamical access is the same to the forward one, then U is of parabolic type. The above domains have a uniformization $\Psi : \mathbb{H} \to U$ and a linear map $G : \mathbb{H} \to \mathbb{H}$ such that $(f|_U) \circ \Psi = \Psi \circ G$ . If U is of hyperbolic type I or II, then G(z) = az, a > 1. For U of parabolic type, then G(z) = z + 1. Thus the diagram bellow commutes. $$\begin{array}{cccc} & & & & & & & & & & & & & & & & &$$ Now, let us denote by $\alpha$ the geodesic $it \in \mathbb{H}$ , t > 0 and by $A_a$ the annulus $\mathbb{H}/G$ with modulus $m(A_a) = \pi/\log(a)$ . Let $c = \alpha/G$ be the core geodesic of $A_a$ , with length $\log(a)$ . The dynamic action of f in U defines the quotient space U/f. Hyperbolic domains of type I and II, have as quotient space U/f, an annulus $A_a$ which is conformally the same that the quotient annulus $\mathbb{H}/G$ .

Definition 2. (Weak hyperbolicity) We say that f is weakly hyperbolic if there are constants r > 0 and $\delta < \infty$ such that, for all $z \in J_f - \{preparabolic points\}$ , there is a subsequence of iterates $\{f^{n_k}\}$ such that $$deg(f^{n_k}: W_k(z) \to D(f^{n_k}(z), r)) \le \delta$$ where $W_k(z)$ is the connected component of $f^{-n_k}(D(f^{n_k}(z),r))$ containing z. Due to the condition of the postcritical set and the Julia set above, our maps do not have parabolic points.

Definition 3. Let $\Lambda$ be a set of (complete and open) geodesics in U, we say that $\Lambda$ is a Baker lamination in U, if the elements in $\Lambda$ satisfies: - (i) It is invariant under the action of f. That is, if $\lambda \in \Lambda$ , then $f^n(\lambda) \in \Lambda$ for any n a positive integer. Also if we consider the inverse branch of $f^{-n}$ from U to U, we have that $f^{-n}(\lambda) \in \Lambda$ . - (ii) For any $\lambda, \lambda' \in \Lambda$ , then $\lambda \cap \lambda' = \emptyset$ . - (iii) Its end points are well defined points in $\partial U$ and any two geodesics have different end points in $\partial U$ . (iv) There are not self accumulating leaves. That is, all geodesics in $\Lambda$ admit a good neighborhood of thickness $\delta$ with the property that good neighborhoods are pairwise disjoint. See figure 2 (page 11), for an example. Let us denote by $End(\gamma)$ the two end points of the geodesic $\gamma$ and by $\bar{\gamma} = \gamma \cup End(\gamma)$ . We observe that condition (iii) can be restated as $End(\gamma) \cap End(\gamma') = \emptyset$ for any $\gamma, \gamma' \in \Lambda$ . If the Baker domain is periodic with $U = \bigcup_{i=1}^{n-1} U_i$ and $fU_i = U_{i+1}$ , then $f^nU_1 = U_1$ and we can consider a Baker lamination $\Lambda_1$ in $U_1$ , invariant under $f^n$ and so $f^k\Lambda_1 = \Lambda_k$ , k = 1, ..., n-1 define a Baker lamination in $U_{k+1}$ respectively, invariant under $f^n$ . The construction defines to $\Lambda = \bigcup \Lambda_{i=1}^n$ as a Baker lamination in the periodic domain U. Good neighborhood in $\Lambda$ are constructed similarly, first for the Baker lamination in $U_1$ and then by iteration for any element of $\Lambda$ . Notation: We will denote by $\mathcal{R}(\Lambda) = \bigcup_k f^{-k}(\Lambda)$ , the full orbit of $\Lambda$ . Let $\overline{\Lambda}$ be the set of geodesics in $\Lambda$ together with its end points and denote by $\mathcal{R}(\overline{\Lambda}) = \bigcup_k f^{-k}(\overline{\Lambda})$ , the full orbit of $\overline{\Lambda}$ . Let us denote by $\mathcal{V}(\Lambda) = \bigcup_k f^{-k}(V_{\delta}(\Psi\Lambda))$ , the full orbit of the good neighborhoods. Let $\mathcal{U}$ the full orbit of the Baker domain U.

Definition 4. (McMullen [Mc1]) J(f) is thin at $\infty$ if for all $z \in \mathbb{C}$ there exist R and $\epsilon$ such that density $J(f), D_r(z) < 1 - \epsilon$ , for all r > R. Where $D_R(z)$ is the disc centered at z with radius R and $density(E,G) := \frac{area(E \cap G)}{area G}$ . areaG. The main task of this section is to prove the following result:

Definition 5. The Hurwitz marked class $\mathcal{M}_f$ of an entire transcendental map f, is the space Top(f) modulo the action of $QC_0(f)$ , i.e. $\mathcal{M}_f = Top(f)/QC_0(f)$ . If $Sing(f^{-1})$ is discrete, $\varphi \in Top(f)$ can be choosen quasiconformal and in this case $\psi$ is also quasiconformal. Eremenko and Lyubich proves in [EL] that if $\phi_0 \circ f = g_0 \circ \psi_0$ and $\phi_1 \circ f = g_1 \circ \psi_1$ and there is an isotopy $\phi_t$ connecting $\phi_0$ with $\phi_1$ fixing the singular values, then $g_0 = g_1$ . Now, consider the surface $S = \mathbb{C} - Sing(f^{-1})$ and let us denote by T(S) the usual Teichmüller space of S. Given a quasiconformal map $\varphi \in \mathcal{M}_f$ , the restriction of $\varphi$ to the surface S gives an element in T(S). Inversely, assuming that the measure of $Sing(f^{-1})$ is zero, given a quasiconformal map $\varphi$ in T(S), it can be extend to a quasiconformal map of $\mathbb{C}$ . As shown in [EL] there is a homeomorphism $\psi$ and an entire transcendental map such that $\varphi \circ f = g \circ \psi$ . This implies that there is a well defined bijective map from $\mathcal{M}_f$ to T(S) which induces a topology in $\mathcal{M}_f$ . For a definition of the Teichmüller space of f, Teich(f) see for instance [McSu], [MNTU], [FH1]. According to the situation we describe above, Teich(f) is included in T(S) as a subspace, hence Teich(f) includes naturally as a subspace of $\mathcal{M}_f$ . Entire Transcendental maps satisfy properties (a) and (b) of Section 4, satisfies that m(J(f)) = 0. According to [FH], we have T eich(f) = T eich(U/f) ×ⁱ T eich(Di/f), being Dⁱ the full orbit of an invariant Fatou domains other than U (there are no deformations in the Julia set). For f with a univalent periodic Baker domain of hyperbolic type I, we have that T eich(f) is infinite dimensional complex Banach space . Infinite dimensional Teichm¨uller spaces have a Teichm¨uller metric which is non-differentiable and has arbitrarily short simple geodesics [EaLi], but are embedded into certain spaces of quadratic differentials by a theorem of Bers [GH]. Many other properties can be found in [NV], [GH] and [GL]. We are interested in the boundary of T eich(f) as subspace of M^f . For instance a consequence of Theorem 1 is that T eich(f) is not compactly contained in M^f . By Theorem 2, the maps which are limits of pinching along Baker laminations other than λ^∞ are in the boundary of T eich(f) and are in M^f . These limits are similar to the cusp groups in the Bers boundary, whose density was proved by McMullen [Mc3] and are also reminiscent of the conjecture in [Mc2], that parabolic periodic domains are dense in the boundary. Although no much is known about the boundary of Teichm¨uller spaces of infinite dimensions, it would be interesting to explore it and a problem is if our pinched limits are dense in the boundary of the slice T eich(U/f). In the next proposition we are interested in the development of U to a periodic domain as (ft) tends to the boundary of T eich(f), when f^t deforms in the slice T eich(U/f), this deformation is not necessarily by pinching a lamination.