Abstract:
In this talk, I will discuss the problem of studying  billiard
trajectories in a polygon whose interior angles are  rational
multiples of pi.  This leads naturally to the study of
translation surfaces, which is another term for the notion  of a
holomorphic 1-form on a (closed) Riemann surface.   The moduli
space of such objects admits an action of  SL(2,R) with a finite
invariant measure that can be used  to give information about the
dynamics of billiard trajectories  in the original polygon; for
example, a theorem of  Kerckhoff-Masur-Smillie tells us that for
almost every  direction theta, every trajectory with initial
direction theta  will be uniformly distributed.  In this talk, I
will describe some  refinements of the general technique of
exploiting dynamics  on the moduli space to obtain information
about dynamics of  billiard trajectories. This talk is intended
for a general audience.

Abstract:

The main objects of interest of this talk are topological
dynamical systems on measure metric spaces that are
equicontinuous with respect to a measure. We will talk about
different notions of continuity and equicontinuity with respect
to a measure and characterize them. We will see how to check for
weak convergence of sequences of measures of a certain family.

Abstract: Recently, Peter Dukes alerted me to a way of computing
the digits of Pi via a "billiard" game. I will use this
remarkable process to introduce the collision transformation,
discuss its properties and provide a geometric proof that the
number of collisions in a system of N hard spheres (any N,
arbitrary diameters and masses) in all space is always finite.
This answers a question originally asked by Sinai.

Abstract: The Krieger generator theorem says that every
invertible measure- preserving system with finite
measure-theoretic entropy can be embedded into a full shift with
strictly greater topological entropy. In joint work with Anthony
Quas, we extend Krieger’s theorem to include toral automorphisms
(which are not necessarily hyperbolic) to give a positive answer
to a question of Lind and Thouvenot.

Abstract: In the 1960's, Stephen Smale initiated an ambitious program to understand the dynamics of a certain class maps on smooth manifolds, which he called Axiom A. David Ruelle gave a definition of a Smale space to describe, in purely topological terms, the dynamics of an Axiom A system on its non-wandering set. I will discuss these definitions and give several concrete examples. Later, Anthony Manning proved that the zeta function for such systems was rational and Rufus Bowen conjectured that this was due to the existence of an underlying homology theory. A partial solution to this was given by Krieger and also Bowen and Franks when they constructed a very beautiful invariant for shifts of finite type. I will discuss this invariant and then show how it can be extended to all Smale spaces as a homology theory, as predicted by Bowen.

An open dynamical system is a measure preserving transformation
on a state space with a 'hole'. As orbits enter the hole, they
are lost from the system. Example: a billiard table.  Typically,
Lebesgue almost every orbit enters the hole eventually, so there
can be no absolutely continuous invariant measure. Instead we
look for a quasi-invariant measure, meaning, a measure stationary
after conditioning on the state of not being in the hole.  In
this setting one has a Perron-Frobenius operator and P-F equation
whose solution gives the density of a quasi-invariant measure.

Borrowing shamelessly from Anthony's careful setup last week in
the closed dynamics setting, I hope to describe the basic
functional analysis setup and a fundamentally different method
for calculating (= approximating) the quasi-invariant density
using optimization instead of discretization and projection.

Keywords:  Open dynamics, Perron-Frobenius operator, density,
maximum entropy principle, moment constraints.

Abstract: Ulam's method is a very successful and practical method
for computing densities of absolutely continuous invariant
measures of transformations. In this talk, we recall our recent
work on Multiplicative ergodic theorems and indicate how this
work can be used to give a version of Ulam's method for
non-autonomous dynamical systems.  (This is joint work with
Gary Froyland and Cecilia Gonzalez-Tokman)

We consider an arbitrary directed graph on finitely many vertices, on which each edge is assigned a positive length and a positive weight.  The asymptotic growth is given for the weighted count of paths of length at most t, as t approaches infinity.