Abstracts, Discrete Methods in Ergodic Theory
Northwestern University
February 2425, 2011

Tim Austin, Brown University
The quantitative ergodic theorem and Banach spaces embeddings
of the Heisenberg group
The classical Mean Ergodic Theorem asserts convergence of
ergodic averages to some invariant function, but gives no effective
control over how fast is this convergence (say, in norm).
Quantitative versions of this theorem turn out to be quite subtle: one
must search ask instead for long `epochs' of time over which the
ergodic averages are approximately constant.
I will show an application of these quantitative results to estimating
the minimal distortion of the word metric on the discrete Heisenberg
group under Lipschitz embeddings into uniformly convex Banach spaces,
where one of the quantitative variants of the ergodic theorem gives
essentially sharp bounds on the `compression exponent' which
quantifies this distortion.
Based on joint work with Assaf Naor and Romain Tessera.

Jon Chaika, University of Chicago
Some results on diophantine approximation for dynamical systems
This talk will present two general results on diophantine approximation
for dynamical systems.
1) Many measured quantities are almost everywhere 0
or infinity (joint with M. Boshernitzan).
2) Orbits of measure preserving maps of [0,1] have badly approximated
points.

Michael Christ, University of California, Berkeley
On the extremizers of a certain L^p norm inequality
Over the last few decades, a vast literature has developed concerning
the mapping properties, in the scale of L^p spaces, of linear integral
operators which
involve singularities and/or curvature. Some of these inequalities
have been established
by exploiting a combinatorial perspective.
There are natural inverse problems: Characterize those functions which
extremize
the inequalities up to bounded factors; determine whether extremizers
exist; identify
these if possible; determine their quantitative and qualitative
properties in cases when
identification is not possible.
I will outline a series of works on these problems, for one of the
most canonical
of the Radonlike transforms. The most recent result is that
extremizers for this particular
inequality are infinitely differentiable. This is established via
analysis of regularity of
solutions of a certain nonlinear EulerLagrange equation. The main
tool is a new family
of weighted L^p norm inequalities. (joint work with Qingying Xue)

Alex Furman , University of Illinois, Chicago
Actions on homogeneous spaces as measureclass preserving systems
Consider actions of a discrete subgroup Gamma of G on a homogeneous space
G/H
as an abstract measure class preserving Gammaaction (we use the
Haar measure class).
It turns out that under mild conditions all measurable equivariant
maps between such actions,
relatively measurepreserving quotients of such systems, joinings, and
quasifactors, are
essentially algebraic. This is a joint work with Uri Bader, Alex
Gorodnik, and Barak Weiss.

Alex Gorodnik, Bristol University
Quantitative ergodic theory and Diophantine approximation
The fundamental problem of Diophantine approximation is to quantify
the density of rational numbers in reals. We investigate this problem
on surfaces equipped with group actions, and establish analogues
of Khinchin and Jarnik theorems in this setting. The main ingredient
of the proof is an ergodic theorem with explicit rate. This is a joint
work with A.Ghosh and A.Nevo.

Mike Hochman, Princeton University
Dimension growth of repeated convolutions of fractal measures
I will discuss the growth of dimension of fractal sets and
measures on R^d under repeated sumset and convolution operations,
respectively. I will explain how suitable local nondegeneracy
assumption imply that the dimension converges to d. The method yields
rates given suitable initial data.

Dima Kleinbock, Brandeis University
Ergodic Theory and Schmidt Games
A game invented by Wolfgang Schmidt in the 1960s turned out to be
a useful tool for studying exceptional sets in ergodic dynamical systems.
I will survey a historical development of this theme and then describe some
recent results, joint with Ryan Broderick, Yann Bugeaud, Lior Fishman and
Barak Weiss.

Hee Oh, Brown University
Counting circles and Ergodic theory of Kleinian groups
We will present recent results on counting and equidistribution
of circles in a given circle packing invariant under a Kleinian group
and discuss how the dynamics of flows on infinite volume hyperbolic
manifolds are related.
Our results apply to Apollonian circle packings, Sierpinski curves and
Schottky dances, etc.

Terry Tao (colloquium), University of California, Los
Angeles
Singlescale ergodic theory
Ergodic theory has traditionally been concerned with the
analysis of averages of dynamical systems in the asymptotic limit when
the scale parameter N goes to infinity. However, more recently some
"singlescale" variants of this theory have begun to emerge, in which
the scale parameter N is large but fixed; such variants are useful in
various finitary applications, for instance in counting patterns such
as arithmetic progressions in the primes. (The recently established
inverse conjecture for the Gowers uniformity norms can be viewed as
the singlescale analogue of the HostKra characteristic factor
theorem in ergodic theory.)
In this singlescale context, one loses the ability to take classical
limits (though one can use ultralimits and nonstandard analysis as a
partial substitute), which deprives the theory of such basic tools as
the ergodic theorem. Nevertheless, a surprising amount of the
asymptotic theory still survives in the singlescale context, though
sometimes with some new features. We illustrate this with the
equidistribution theory of linear or polynomial sequences in tori or
nilmanifolds, and with a number of other examples.

Terry Tao, University of California, Los Angeles
The inverse conjecture for
the Gowers uniformity norms on [N]
The Gowers uniformity norms U^{k+1}[N], introduced by
Gowers in 2001, is very useful when counting the number of linear
patterns (such as arithmetic progressions of length k+2) in a set of
integers. The inverse conjecture gives a necessary and sufficient
condition for the Gowers norm of a bounded function to be large;
roughly speaking, a function n \mapsto f(n) has large U^{k+1}[N]
norm if and only if it correlates with a bounded complexity
nilsequence n > F( g(n) \Gamma ) on a nilmanifold G/\Gamma
of step (or degree) k. This generalises the Fourieranalytic case
k=1, in which a function has large U^2[N] norm if and only if it
correlates with a Fourier phase n > e(\xi n); this case is
ultimately behind the Fourieranalytic proofs of results such as
Roth's theorem, and the inverse conjecture for higher k can
similarly be used to give a proof of Szemeredi's theorem.
In this talk we outline a proof of the general case of the inverse
conjecture, in joint work with Ben Green and Tamar Ziegler. The
arguments are based on those of Gowers, with the main new difficulty
being that of "integrating" a family of degree k1 nilsequences to
obtain a single degree k nilsequence as an "antiderivative"