Abstracts, Discrete Methods in Ergodic Theory

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 Radon-like 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 Euler-Lagrange 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 measure-class preserving systems
Consider actions of a discrete subgroup Gamma of G on a homogeneous space G/H as an abstract measure class preserving Gamma-action (we use the Haar measure class). It turns out that under mild conditions all measurable equivariant maps between such actions, relatively measure-preserving quotients of such systems, joinings, and quasi-factors, 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 non-degeneracy 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
Single-scale 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 "single-scale" 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 single-scale analogue of the Host-Kra characteristic factor theorem in ergodic theory.) In this single-scale 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 single-scale 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 Fourier-analytic 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 Fourier-analytic 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 k-1 nilsequences to obtain a single degree k nilsequence as an "antiderivative"