Complex Analysis and Dynamics Seminar

Spring 2007 Schedule

Feb. 2: Linda Keen (Lehman College and GC, CUNY)
Siegel Disks for a Family of Entire Functions
In a joint work with Gaofei Zhang, we show that functions in the
family f_{a}(z) = (m z + a z^{2}) e^{z}, with
the complex parameter a nonzero,
m = exp(2 \pi i t) and t irrational
of bounded type, have Siegel disks whose boundary is a quasicircle
containing one or both of the critical points.

Feb. 9: Saul Schleimer (Rutgers University)
Covers and the Curve Complex
A covering map between surfaces induces an inclusion of curve
complexes, by taking preimages of curves. We prove that this map is a
quasiisometric embedding. The result depends on a delicate interplay
between the geometry of Teichmuller space and of the curve complex. This
is joint work with Kasra Rafi.

Feb. 16: Ed Taylor (Wesleyan University)
Quasiconformal Homogeneity of Hyperbolic
Manifolds
A complete hyperbolic manifold is uniformly quasiconformally
homogeneous if there exists a K so that every pair of
points on the manifold can be paired by a
Kquasiconformal automorphism of the manifold. In
dimensions three and above, there exists a fairly complete geometric and
topological description of uniformly
quasiconformally homogeneous hyperbolic manifolds. However, in dimension
two the situation is more wide open.
This is the first of a twopart talk. I will introduce the basics
in the study of the quasiconformal homogeneity of
hyperbolic manifolds, as well as recent work on the
quasiconformal homogeneity of hyperbolic surfaces having
a "sufficiently large" conformal automorphism group.
On March 30th, Petra BonfertTaylor will give a
second talk in which certain extremal problems in the study of
quasiconformal homogeneity will be of focus.

Feb. 23: Huyi Hu (Michigan State University)
Equilibriums of Some NonHolder Potentials
Consider a onesided subshift of finite type with some
potential function which satisfies the Holder conditions everywhere
except possibly at a fixed point and its preimages.
We discuss the properties of the equilibriums of such potentials.
This includes the convergence rate of the tails, finiteness, exactness,
Gibbs properties, and uniqueness.

Mar. 2: Paul Norbury (Boston University)
WeilPetersson Volumes and Cone Surfaces
The moduli space of hyperbolic surfaces of genus g with n
geodesic boundary
components is naturally a symplectic manifold and hence has a welldefined
volume. Mirzakhani proved that the volume is a polynomial in the lengths of the
boundaries by computing the volumes recursively in g and n.
By allowing cone
angles on hyperbolic surfaces, we give new recursion relations between the
volume polynomials. This has interesting consequences for the geometry of the
moduli space.

Mar. 9 and 16: No meeting

Mar. 23: Ara Basmajian (Hunter College, CUNY)
Convergence Groups and Isometries of Hyperbolic Spaces
The isometry group of (real, complex, quaternionic, or Cayley)
hyperbolic space extends naturally to a group of conformal homeomorphisms
of the boundary sphere. In joint work with Mahmoud Zeinalian, we show that
this extension is a maximal convergence group. Moreover, we show that any
family of uniformly quasiconformal homeomorphisms has the convergence
property.
After defining convergence groups and discussing their basic properties,
we will indicate the proofs of the above theorems as well as some of
their consequences.

Mar. 30: Petra BonfertTaylor (Wesleyan University)
Quasiconformal Homogeneity, Part II
This is a continuation of the talk "Quasiconformal
homogeneity of hyperbolic manifolds", given by Edward Taylor on
February 16th.
Recall that a complete hyperbolic manifold is uniformly
quasiconformally homogeneous if there exists a K so that every
pair of points on the manifold can be paired by a Kquasiconformal
automorphism of the manifold. We will review the definition, some
basic results and some geometric and topological constraints associated
with quasiconformal homogeneity. We will then focus on quasiconformal
homogeneity in dimension 2, which has to be dealt with quite
differently from the higherdimensional case. We will present recent
results for hyperelliptic surfaces and planar domains.

Apr. 13: Christian Wolf (Wichita State University)
Natural Invariant Measures for Hyperbolic and Parabolic
Dynamical Systems
In this talk I will introduce several classes of natural
invariant measures including generalized physical and SRBmeasures
and measures of maximal dimension. These measures are associated with
the "typical" dynamics of the underlying dynamical system. I will then
present some recent results concerning the existence, uniqueness and
characterization of these measures for hyperbolic and parabolic
systems. Even though the presented material is technical in
nature, I will highlight ideas rather than technical details in
order to make the talk accessible to a general audience.

Apr. 20: No meeting

Apr. 27: Andrew Haas (University of Connecticut)
Cusp Excursions and Metric Diophantine Approximation for Fuchsian Groups
In this talk we shall derive several results that describe the rate
at which a geodesic makes excursions into and out of a cusp on a
finite area hyperbolic surface and relate them to approximation with
respect to the orbit of infinity for an associated Fuchsian group.
This provides proofs of several important theorems from metric
diophantine approximation in the context of Fuchsian groups. It also
illuminates the classical theory and produces new estimate on the
rate of growth of the partial quotients in the continued fraction
expansion of a generic real number.

May 11: Jane Gilman (Rutgers University)
Finding Parabolic Dust and the Structure of TwoParabolic Space
Every nonelementary subgroup of PSL(2,C) generated by two
parabolic transformations is determined up to conjugacy by a nonzero
complex number, and each nonzero complex number determines a twoparabolic
generator group. I will discuss a structure theorem for twoparabolic
space, that is, the full representation space modulo conjugacy of
nonelementary twoparabolic generator groups. Portions of twoparabolic
space have been studied by many authors including Bamberg, Beardon,
LyndonUllman, KeenSeries, Minsky, Riley, Wright, and GilmanWaterman.
These include portions corresponding to free and nonfree groups, discrete
and nondiscrete groups, manifold and orbifold groups, Riley groups, NSDC
groups and classical and nonclassical Schottky groups. I will discuss
locating nonfree parabolic dust, that is, groups with
additional parabolics, by iterating the GilmanWaterman classical
tangent Schottky boundary.
