For more information about this meeting, contact Dmitri Burago.
| Title: | Metastability in ergodic theory |
|---|
| Seminar: | Center for Dynamics and Geometry Seminar |
|---|
| Speaker: | Jeremy Avigad, Carnegie Mellon |
|---|
| Abstract: |
|---|
| In dynamical systems and ergodic theory, one tries to characterize the
behavior of systems that evolve over time. These fields, however,
often use methods that are difficult to interpret in computable or
quantitative terms.
For example, let $T$ be a measure-preserving transformation of a space
$(X, \mathcal B, \mu)$, let $f$ be a measurable function from $X$ to
$\mathbb{R}$, and for every $x \in X$ and $n \in \mathbb{N}$ let $(A_n
f)x = (f x + f(T x) + \ldots + f(T^{n-1} x)) / n$. The pointwise
ergodic theorem says that this sequence of averages converges for
almost every $x$, and the mean ergodic theorem says that the sequence
$(A_n f)$ converges in the $L^2$ norm. But, in general, one cannot
compute a rate of convergence from the initial data. Describing joint
work with Philipp Gerhardy and Henry Townser, I will explain how
proof-theoretic methods provide classically equivalent formulations of
the ergodic theorems which are computably valid, and yield additional
information.
Roughly, the additional information amounts to quantitative bounds on
how long one has to wait to find long pockets of stability in the
sequence of ergodic averages. These bounds are, moreover, independent
of the measure-preserving system. The fact that such bounds can often
be obtained is an instance of a phenomena that Terence Tao has called
``metastability.'' As a second example, I will discuss the
Furstenberg-Zimmer structure theorem, which shows that any measure
preserving system can be decomposed into to a transfinite sequence of
compact extensions terminating with the maximal distal factor. I will
describe a metastable version of this theorem, obtained jointly with
Towsner, which shows that factors approximating the behavior of the
maximal distal factor can occur much earlier in the sequence. |
Room Reservation Information
| Room Number: | MB106 |
|---|
| Date: | 11 / 17 / 2010 |
|---|
| Time: | 03:35pm - 05:05pm |
|---|
