Saturday, November 5, 2005
Schedule of Events
All talks will be in Old Main (2nd floor)
600 Lincoln Avenue, Charleston, IL
Full Schedule information will be released at a later
time.
9:00 AM Coffee and Refreshments
9:30 AM
Speaker : Renato Feres, Washington University
Title: Random walks and diffusion derived from
billiards
Abstract: We introduce a class of random
dynamical systems derived from billiard maps, which we call random
billiards, and study certain random walks on the real line obtained
from them. The interplay between the billiard geometry and the
stochastic properties of the random billiard is investigated. Our
main results are concerned with a description of the spectrum of the
random billiard's Markov operator and with properties of a diffusion
limit under appropriate scaling.
10:30 AM
Speaker: Gregory Galperin, Eastern Illinois University
Title:
Generalized billiards inside an infinite strip with periodic laws of
reflection along the
strip's boundaries.
ABSTRACT: The speaker will give a constructive description of
generalized billiards inside
an infinite strip with a periodic law of reflection on each of the two (top, bottom) boundaries. Each
boundary is equipped with a periodic
lattice, where the number of lattice's nodes between any two successive reflection points can be
arbitrary. The speaker will present a full description of the structure of the
set of billiard trajectories for such billiards, explain why the spatial
chaos exists in such systems, and will find the exact value of the spatial
entropy in the class of monotonic billiard trajectories.
11:30 am
Speaker: Richard
Bishop, University of Illinois
at Urbana-Champaign
Title: Caustics of Circular Billiards
Abstract: The
qualitative and quantitative properties of the caustics
of a circular mirror are described. For a point source, the n th
caustic is the envelope of the family of lines containing the
n+1 st reflective segment. It is an analytic curve with 4 cusps and may
extend outside the circle, depending on n and the distance of the
source from the center. If we view the paths as geodesics of a
double disk, then the n th conjugate locus of the source consists of
arcs of one or two of the caustics. The simplest of these is the first
conjugate and caustic locus of a point on the bounding circle, which
was already identified by Huygens as a cardioid.
2:00 PM
Speaker: Keith Burns, Northwestern University
Title: Constructing
metrics with ergodic geodesic flow in dimension three
This talk will describe a
joint project with Marlies Gerber to construct
metrics with egodic geodesic flow on three dimensional manifolds. The
previously known method of obtaining such metric depends on deep results
of Thurston and others that allow one to find inside any three manifold
a
knot whose complement has a hyperbolic structure. In contrast, the
starting point of our approach is a triangulation of the manifold. We
introducte negative curvature by turning the 3-simplices of the
triangulation into an ideal hyperbolic tetrahedra. Then we smooth out
the
singularities along the edges and at the vertices in such a way that the
positive curvature which are forced to introduce does not damage the
good
behaviour of the geodesic flow produced by the negative curvature.
3:30 PM
Speaker: Victor Donnay, Bryn Mawr College
Title:
Billiards: Can one prove ergodicity when both defocusing and pure
divergence are present?
Abstract: In the Sinai billiard collisions with the concave
boundaries cause pure divergence of families of trajectories. In
the Bunimovich Stadium (and related billiards), collisions with the
convex boundary cause converge which is then followed by focusing and
divergence (termed defocusing). Thus there are two different mechanisms
known to produce chaotic motion in billiards. What happens if
both behaviors are present in the same system?
We designate systems in which the two behaviors are present as "partial
focusing" and argue that showing ergodicity or positive measure entropy
for such systems is a very delicate issue.
Specifically, we examine partially focusing system that arise in
geodesic flow systems (surfaces with negative curvature and "partially
focusing caps"). In earlier work, we constructed surfaces with
"focusing caps" whose geodesic flow was ergodic due to the defocusing
mechanism. Here we show that there are partially focusing systems,
arbitrarily close to the focusing ones, that are not ergodic (they
contain stable, elliptic periodic orbits).
The moral is that if within a system one finds both defocusing and pure
divergence, and within the system one can move continuously from the
one behavior to the other, then there is always the risk of having
elliptic orbits.
Visitors may park at Parking lot X
There will be breaks between the talks. Lunch will be
taken around 12:30 pm.
Suggested Accommodations:
(Charleston,IL) BestWestern 1-800-528-8161
(Charleston, IL) Econo Lodge 1-800-424-4777
(Charleston, IL) Queen Anne's (B and B) 217-345-1288
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