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 orientationpreserving branched coverings of the 2sphere.
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 criticallyfinite branched coverings and subhyperbolic semirational 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 2parameter 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 nonperiodic 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 finitedegree 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 infinitedegree 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.

Sep 30: No meeting

Oct 7: No meeting

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 (JenkinsStrebel) differential
with characteristic ring domains $D_j\{p: z_j(p)<\epsilon\}$, where the domains $D_j$ are nonoverlapping 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 nonoverlappoing 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 nonoverlapping 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) 243257), 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 infinitedimensional parameter spaces. We will also discuss the intimate relationship
between extending holomorphic motions (over infinitedimensional 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
infinitedimensional 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 measuretheoretic 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
EarleIvanovKraMarkovicRoyden'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 nonempty 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 KleinMaskit 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 UrbanaChampaign)
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:
1:452:45 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 realanalytic
and strictly positive function on the interior of the barycenter set.
2:503:50 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 socalled 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 oneparameter 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, BowditchEpstein and Penner’s cell decomposition of the decorated Teichmüller space is reproduced.
