Complex Analysis and Dynamics Seminar

Fall 2006 Schedule

Sept. 8: Hrant Hakobyan (Stony Brook University)
Conformal Dimension of Cantor Sets

A subset of the line is quasisymmetrically (qs) thick if its image under every qs self-map of the line has positive length. The conformal dimension of a metric space is the infimal Hausdorff dimension of all its qs images into metric spaces. Bishop and Tyson asked if there is a subset of the line which is not qs thick but has conformal dimension 1. We answer this affirmatively by showing that middle-interval Cantor sets are minimal for conformal dimension if they have Hausdorff dimension 1. The same result holds for products of these Cantor sets.

Sept. 15: Yunping Jiang (Queens College and GC, CUNY)
Asymptotic Geometry and Hausdorff Dimension of a Family of Asymptotically Non-Hyperbolic Cantor Systems

A family of Cantor systems associated with a lower smoothness family of folding maps is defined and studied. Such a family often arises in dynamical systems as hyperbolicity is created. We study the asymptotic geometry of the family of Cantor systems and use it to study asymptotically Hausdorff dimension. We show that the bridge geometry of the family of Cantor systems is uniformly bounded and that the gap geometry as well as the Hausdorff dimension of the family is regulated by the size of the leading gap.

Sept. 22: No meeting

Sept. 29: Frederick Gardiner (Brooklyn College and GC, CUNY)
Teichmuller Theory for the Associahedron

The associahedron of dimension n is a simplicial complex built according to rules indicated by the different associations of n+2 letters. We explain how its structure is connected to four basic theorems in Teichmuller theory.

Oct. 6: Michael Yompolsky (University of Toronto)
New Results on Computability of Julia Sets

Informally, a planar set is computable if there exists an algorithm to draw its picture on a computer screen with an arbitrary pixel size. This talk will present new results obtained jointly with Mark Braverman which settle several open questions on computability of Julia sets.

Oct. 13: Kourosh Tavakoli (Lehman College, CUNY)
Iterated Holomorphic Function Systems

Consider all the backward iterated function systems corresponding to the sequences of holomorphic functions from the unit disk D into a subdomain X. Lorentzen and Gill showed that if X is relatively compact in D, then every iterated function system has a unique limit function which is a constant inside X. In other words, they showed that relative non-compactness of X is necessary in order to have a boundary point as a limit function. Keen and Lakic used the notion of hyperbolic Bloch domain, first introduced by Beardon et al., and showed that if X is not Bloch in D, every boundary point of X is a limit function of some iterated function system. In this talk we generalize this result and show that relative non-compactness of X in D is a sufficient condition to have a boundary point as a limit function.

Oct. 20: Jun Hu (Brooklyn College and GC, CUNY)
Associahedra and Spaces of Finite Earthquake Maps

This is a continuation of Fred Gardiner's talk on Teichmuller theory of the associahedron, but I will present it as an independent talk by focusing on the connection of the associahedra to the spaces of finite earthquake maps or measures. We show that the associhedra give nice geometric visualizations of the compactifications of the real Teichmuller spaces of finite points arranged on the unit circle.

Oct. 27: Howard Masur (University of Illinois at Chicago)
Dynamics of Flows on Translation Surfaces

Suppose one has a polygon in the plane such that for every side there is a parallel side of the same length. Gluing the sides together by a parallel translation gives a translation surface. Equivalently, such a structure is given by an Abelian differential on a compact Riemann surface. For each direction theta, there is a flow by straight lines in that direction. I will discuss what is known about these flows.

Nov. 3: No meeting

Nov. 10: Reza Chamanara (Stony Brook University)
Convex Ideal Hyperbolic Polyhedra and Bending Invariants of Jordan Curves

It is known from the work of Igor Rivin that convex ideal hyperbolic polyhedra are uniquely determined up to isometries by their combinatorial pattern and dihedral angles at all edges. I will give a short proof of this theorem and discuss how it can help us better understand the invariants of Jordan curves defined by their convex hulls in the 3 dimensional hyperbolic space.

Nov. 17: Anca Radulescu (University of Colorado)
Complexity and Entropy Computability and a Model for Schizophrenia

Evaluating the complexity of a dynamical system and following its evolution under perturbations has been a subject of research in recent years. To give a partial answer to the question of "when is entropy effectively computable," I will first describe a few distinct algorithms currently used for one and two-dimensional maps, with their advantages and fall-backs. I will then try to illustrate the importance of possessing such complexity estimation methods. I will describe their potentially priceless impact when dealing with a clinical model: the neurodegenerative system that we believe constitutes the trigger for schizoprenic symptoms.

Dec. 1: Jeff McGowan (Central Connecticut State University)
Lengths of Closed Geodesics on Random Riemann Surfaces

Short geodesics are important in the study of the geometry and spectra of Riemann surfaces. Bers' theorem gives a global bound on the length of the first 3g-3 geodesics. We use Brooks and Makover's construction of random Riemann surfaces to investigate the distribution of short (i.e., length < log g) geodesics on random Riemann surfaces. We calculate the expected value of the shortest geodesic, and show that if one orders prime non-intersecting geodesics by length l1 =< l2 =< ... =< lk =< ..., then for fixed k, if one allows the genus to go to infinity, the expected value of lk is independent of the genus. We will also discuss some preliminary results on nice homology bases for random Riemann surfaces.

Special Seminar, Dec. 8, 11:30 am - 12:30 pm, in room 4419:

Marco Lenci (University of Bologna, Italy)
The (Aperiodic) Lorentz Gas and its Recurrence

A Lorentz gas is the dynamical system of a point particle, in the plane or space, that collides elastically with a fixed array of convex "scatterers." The scatterer configuration need not be periodic. This system, or at least its 2D version, is the generalization of the original Sinai billiard to the realm of dynamical systems preserving an infinite measure. I will discuss what is known about this system's most basic properties: hyperbolicity, ergodicity and recurrence. We will see how recurrence (which is not guaranteed, as the Poincare' Recurrence Theorem does not hold for infinite measures) is still largely an open problem.

Dec. 8: Kasra Rafi (University of Connecticut)
Closed Geodesics in the Thin Part of the Moduli Space

In his thesis, Margulis used the mixing properties of the geodesic flow in hyperbolic manifolds to count the number of closed geodesics whose lengths are less than a given constant. A similar program exists for counting the number of closed geodesics in the moduli space of a surface. One difficulty arises from the fact that in the moduli space, unlike a hyperbolic manifold, for every compact set there are closed geodesics that are disjoint from that compact set. We will attempt to address this difficulty.

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