# Complex Analysis and Dynamics Seminar

## Fall 2011 Schedule

#### Sep 9: Yunping Jiang (Queens College and Graduate Center of CUNY) Bounded Geometry and Characterization of Holomorphic Dynamics

In this talk I will define bounded geometry for some orientation-preserving branched coverings of the 2-sphere. It is a useful analytic condition in characterizing holomorphic dynamics by the iteration method on Teichmuller spaces of closed subsets of the Riemann sphere. I will show how to connect bounded geometry with Thurston's topological condition and the canonical condition for critically-finite branched coverings and sub-hyperbolic semi-rational branched coverings.

#### Sep 16: Patrick Hooper (City College of CUNY) Renormalization of Rectangle Exchange Maps Arising from Corner Percolation

I will describe a construction of a 2-parameter family of rectangle exchange maps. I will give an elementary description of a renormalization operation which acts on this family. These rectangle exchange maps arise in connection to the corner percolation model introduced by Balint Toth and studied in detail by Gabor Pete. Because of this construction, there are topological reasons to expect most points to be periodic. The renormalization operation can be used to prove that for almost every member of the family, the rectangle exchange is periodic almost everywhere. In contrast, there are rectangle exchange maps in this family which admit a positive measure set of non-periodic points.

#### Sep 23: Linda Keen (Lehman College and Graduate Center of CUNY) Bounded Geometry and Families of Meromorphic Functions

In 1986 Thurston gave a topological characterization of when it is possible to realize, by combinatorial equivalence, a given finite-degree branched covering map of the sphere with finite postcritical set as rational map. The finiteness of the degree was crucial to his proof. Recently, Hubbard, Schleicher and Shishikura extended his theorem to the famliy of exponential maps. In joint work with Tao Chen and Yunping Jiang, we prove an analog of Thurston's theorem for more general infinite-degree covering maps with certain finiteness properties. We give an analytic condition called "bounded geometry" that characterizes when such a map can be realized by an entire or meromorphic function. In this talk, we will explain our theorem for the family of meromorphic functions with two asymptotic values and no critical points.

#### Oct 14: Frederick Gardiner (Brooklyn College and Graduate Center of CUNY) Moduli for Moving Points

Let $\gamma_1,\ldots,\gamma_n$ be the boundaries of conformal discs centered at points $p_j$ on a Riemann surface $R$, let $V=(v_1,\ldots,v_n)$ be an $n$-tuple of tangent vectors at these points, and let $z_j$ be local parameters on $R$ vanishing at $p_j$ normalized so that $dz_j(v_j)=1.$ The Strebel moduli theorem implies there is a cylindrical (Jenkins-Strebel) differential with characteristic ring domains $D_j-\{p: |z_j(p)|<\epsilon\}$, where the domains $D_j$ are non-overlapping conformal discs in $R$ containing the points $p_j$ that realize the extremum for the extremal problem $$Q(V)= \lim_{\epsilon \rightarrow 0} \ \sup_{\tilde{D_j}} \ \min_j \{ \text{mod}( D_j - \{p: |z_j(p)|<\epsilon\} ) \} + (1/2\pi) \, \log \epsilon.$$ Here the minimum is over $j$ and the supremum is over all families of non-overlappoing conformal discs $D_j$ with the prescribed properties. Another theorem states that there is a unique solution to the following extremal problem: $$U(V)= \inf_{f_j} \ \max_{v_j \neq 0} \{u_j: u_j \, df_j(1)=v_j \},$$ where the $u_j$ are positive numbers, the maximum is over the set of $j$ such that $v_j$ is not equal to zero, and the infimum is over the families of univalent functions $f_j$ mapping the unit disc $\Delta$ into $R$ with non-overlapping images, with $f_j(0)=p_j$ and with $u_j \, df_j (1)=v_j.$ We show that $$U(V) = \exp(-2 \pi Q(V))$$ and explore the consequences.

#### Oct 21: Sudeb Mitra (Queens College and Graduate Center of CUNY) Barycentric Sections and Holomorphic Motions

In their famous paper Extending holomorphic motions (Acta Math. 157 (1986) 243-257), Sullivan and Thurston asked two important questions on extending holomorphic motions over the open unit disk. In this talk, we will discuss these questions over infinite-dimensional parameter spaces. We will also discuss the intimate relationship between extending holomorphic motions (over infinite-dimensional parameter spaces) and lifting holomorphic maps into some appropriate Teichmüller spaces. The central theme is to study barycentric sections for some generalized Teichmüller spaces. We will outline how barycentric sections can be used to obtain results in holomorphic motions over infinite-dimensional parameter spaces. If time permits, we will also discuss what Sullivan and Thurston called the holomorphic axiom of choice and show a simple example where the holomorphic axiom of choice fails. This is the first of a series of talks that I plan to give on the many interesting links between barycentric sections, and quasiconformal and holomorphic motions. It will be based on some recent (and ongoing) joint work with Cliff Earle, Yunping Jiang, Hiroshige Shiga, and Zhe Wang.

#### Oct 28: Saeed Zakeri (Queens College and Graduate Center of CUNY) Conformal Fitness and Uniformization of Holomorphically Moving Disks

Let $\{ U_t \}_{t \in {\mathbb D}}$ be a family of topological disks on the Riemann sphere containing $0$ whose boundaries undergo a holomorphic motion $\partial U_0 \to \partial U_t$ over the unit disk $\mathbb D$. We address the question of when there exists a family of Riemann maps $({\mathbb D},0) \to (U_t,0)$ which depends holomorphically on $t$. We give six equivalent conditions which provide analytic, dynamical and measure-theoretic characterizations for the existence of such family, and explore the consequences. Somewhat curiously, one of these equivalent conditions is the harmonicity of the map $t \mapsto \log \, r_t$ in $\mathbb D$, where $r_t$ is the conformal radius of the pointed disk $(U_t,0)$.

#### Nov. 4: Hideki Miyachi (Osaka University) Unification of the Extremal Length Geometry on Teichmüller Space via Intersection Number

In this talk, I will give a relation between the Gromov product with respect to the Teichmüller distance and the intersection number function on the space of measured foliations via extremal length geometry on Teichmüller space. As an application, I give an alternate approach to Earle-Ivanov-Kra-Markovic-Royden's characterization of isometries on Teichmüller space. Namely, with few exceptions, the isometry group of Teichmüller space with respect to the Teichmüller distance is canonically isomorphic to the extended mapping class group.

#### Nov. 11: Michael Shub (Graduate Center of CUNY) Solving One Homogeneous Polynomial in Two Complex Variables

In 1980, Smale studied the solutions of one complex polynomial equation in one variable. When we adapt his technique to a natural generalization for homogeneous polynomials, many new open problems are encountered on the Riemann sphere. I will discuss the problems and their application to the complexity of equation solving.

#### Nov. 18: Ara Basmajian (Hunter College and Graduate Center of CUNY) Constructing Conformally Scattered Sets

A set is said to be scattered if every non-empty subset has isolated points. Using a recursively defined sequence of derived sets, Ethan Akin showed that for each countable ordinal there exists a compact countable scattered subset of the Cantor set. Moreover these sets are topologically distinct.

In this note, we investigate this problem in the category of conformal mappings. Our interest is in generating scattered sets in the unit circle as orbits of conformal mappings. More precisely, we construct a transfinite sequence of increasing subsets where each subset is a conformal scattering of the previous one- a scattered set is said to be a conformal scattering if it is the closure of the $\langle g \rangle$-orbit of a compact set for some Mobius transformation $g$ preserving the unit disc.

We provide two constructions of such collections of subsets. The first construction uses a transfinite version of the Klein-Maskit combination theorem. In the second construction our sets are all contained in the $\operatorname{SL}(2,\mathbb{Z})$ orbit of $\infty$.

#### Special seminar on Wednesday Nov. 23, 2:30 - 4:00pm, Room 4419 Sergiy Merenkov (University of Illinois at Urbana-Champaign) Quasisymmetric Uniformization of Surfaces

I will discuss recent results on quasisymmetric uniformization of metric surfaces. I plan to mainly emphasize our recent joint work with K. Wildrick on quasisymmetric Koebe uniformization, i.e., a uniformization by circle domains in the sphere.

#### Dec. 2: The seminar will feature two talks: 1:45-2:45 Christian Wolf (City College of CUNY) On Barycenter Entropy for Rational Maps

For a rational map $f$ on the Riemann sphere we study the entropy $H(w)$ of points $w$ in the barycenter set $\Omega(f)$. We show that this entropy is entirely determined by the growth rate of those repelling periodic orbits whose barycenters are close to $w$ and that exhibit sufficient expansion. Assuming additionally that $f$ is hyperbolic, we prove that $H(w)$ is a real-analytic and strictly positive function on the interior of the barycenter set.

#### 2:50-3:50 Guowu Yao (Tsinghua University, visiting Harvard) Geodesic Geometry in the Asymptotic Teichmüller Space

We will talk about the geodesic geometry in an asymptotic Teichmüller space. In contrast with the usual Teichmüller space, in the asymptotic Teichmuller space,
(1) for points in an open dense subset, there are infinitely many geodesics connecting these points to the basepoint;
(2) there are infinitely many straight lines passing through given two points;
(3) there are infinitely many geodesic disks containing given two points.
As for (1), we have some difficulty in dealing with a class of so-called substantial points. Nevertheless, we give an example to show that there are infinitely many geodesics connecting certain substantial points to the basepoint.

#### Dec. 9: Tian Yang (Rutgers University at New Brunswick) A Deformation of Penner’s Simplicial Coordinate

The decorated Teichmüller space of a punctured surface was introduced by R. Penner as a fiber bundle over the Teichmüller space of hyperbolic metrics with cusp ends. To give a cell decomposition of this space, Penner defined the simplicial coordinate $\Psi$ in which the cells can be easily described. As a counterpart of the simplicial coordinate $\Psi$, F. Luo introduced a coordinate $\Psi_0$ of the Teichmüller space of a surface with geodesic boundary, and deformed it to a one-parameter family of coordinates $\{ \Psi_h \}_{h \in {\mathbb R}}$. He and R. Guo also described the images of $\Psi_h$ which turn out to be explicit open polytopes. It is then natural to ask if there exists a corresponding deformation of Penner’s simplicial coordinate $\Psi$. The main result that I will be talking about is an affirmative answer to this question. As an application, Bowditch-Epstein and Penner’s cell decomposition of the decorated Teichmüller space is reproduced.
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