Complex Analysis and Dynamics Seminar
Fall 2006 Schedule
Sept. 8: Hrant Hakobyan (Stony Brook University)
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.
Conformal Dimension of Cantor Sets
Sept. 15: Yunping Jiang (Queens College and GC, CUNY)
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
Asymptotic Geometry and Hausdorff Dimension of a Family of
Asymptotically Non-Hyperbolic Cantor Systems
Sept. 22: No meeting
Sept. 29: Frederick Gardiner (Brooklyn College and GC, CUNY)
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.
Teichmuller Theory for the Associahedron
Oct. 6: Michael Yompolsky (University of Toronto)
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.
New Results on Computability of Julia Sets
Oct. 13: Kourosh Tavakoli (Lehman College, CUNY)
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.
Iterated Holomorphic Function Systems
Oct. 20: Jun Hu (Brooklyn College and GC, CUNY)
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.
Associahedra and Spaces of Finite Earthquake Maps
Oct. 27: Howard Masur (University of Illinois at Chicago)
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.
Dynamics of Flows on Translation Surfaces
Nov. 3: No meeting
Nov. 10: Reza Chamanara (Stony Brook University)
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.
Convex Ideal Hyperbolic Polyhedra and Bending Invariants of Jordan
Nov. 17: Anca Radulescu (University of Colorado)
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.
Complexity and Entropy Computability and a Model for Schizophrenia
Dec. 1: Jeff McGowan (Central Connecticut State University)
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.
Lengths of Closed Geodesics on Random Riemann Surfaces
Special Seminar, Dec. 8, 11:30 am - 12:30 pm, in room 4419:
Marco Lenci (University of Bologna, Italy)
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.
The (Aperiodic) Lorentz Gas and its Recurrence
Dec. 8: Kasra Rafi (University of Connecticut)
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.
Closed Geodesics in the Thin Part of the Moduli Space