Complex Analysis and Dynamics Seminar
Spring 2007 Schedule
Feb. 2: Linda Keen (Lehman College and GC, CUNY)
In a joint work with Gaofei Zhang, we show that functions in the
family fa(z) = (m z + a z2) ez, with
the complex parameter a non-zero,
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.
Siegel Disks for a Family of Entire Functions
Feb. 9: Saul Schleimer (Rutgers University)
A covering map between surfaces induces an inclusion of curve
complexes, by taking preimages of curves. We prove that this map is a
quasi-isometric 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.
Covers and the Curve Complex
Feb. 16: Ed Taylor (Wesleyan University)
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
K-quasiconformal 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.
Quasiconformal Homogeneity of Hyperbolic
This is the first of a two-part 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 Bonfert-Taylor 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)
Consider a one-sided 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.
Equilibriums of Some Non-Holder Potentials
Mar. 2: Paul Norbury (Boston University)
The moduli space of hyperbolic surfaces of genus g with n
components is naturally a symplectic manifold and hence has a well-defined
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
Weil-Petersson Volumes and Cone Surfaces
Mar. 9 and 16: No meeting
Mar. 23: Ara Basmajian (Hunter College, CUNY)
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
Convergence Groups and Isometries of Hyperbolic Spaces
After defining convergence groups and discussing their basic properties,
we will indicate the proofs of the above theorems as well as some of
Mar. 30: Petra Bonfert-Taylor (Wesleyan University)
This is a continuation of the talk "Quasiconformal
homogeneity of hyperbolic manifolds", given by Edward Taylor on
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 K-quasiconformal
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 higher-dimensional case. We will present recent
results for hyperelliptic surfaces and planar domains.
Quasiconformal Homogeneity, Part II
Apr. 13: Christian Wolf (Wichita State University)
In this talk I will introduce several classes of natural
invariant measures including generalized physical and SRB-measures
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.
Natural Invariant Measures for Hyperbolic and Parabolic
Apr. 20: No meeting
Apr. 27: Andrew Haas (University of Connecticut)
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.
Cusp Excursions and Metric Diophantine Approximation for Fuchsian Groups
May 11: Jane Gilman (Rutgers University)
Every non-elementary subgroup of PSL(2,C) generated by two
parabolic transformations is determined up to conjugacy by a non-zero
complex number, and each non-zero complex number determines a two-parabolic
generator group. I will discuss a structure theorem for two-parabolic
space, that is, the full representation space modulo conjugacy of
non-elementary two-parabolic generator groups. Portions of two-parabolic
space have been studied by many authors including Bamberg, Beardon,
Lyndon-Ullman, Keen-Series, Minsky, Riley, Wright, and Gilman-Waterman.
These include portions corresponding to free and non-free groups, discrete
and non-discrete groups, manifold and orbifold groups, Riley groups, NSDC
groups and classical and non-classical Schottky groups. I will discuss
locating non-free parabolic dust, that is, groups with
additional parabolics, by iterating the Gilman-Waterman classical
tangent Schottky boundary.
Finding Parabolic Dust and the Structure of Two-Parabolic Space