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 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.

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 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.

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 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.
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)
Equilibriums of Some Non-Holder Potentials

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.

Mar. 2: Paul Norbury (Boston University)
Weil-Petersson 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 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 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 Bonfert-Taylor (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 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.

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 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.

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 Two-Parabolic Space

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.

Back to the seminar page