# Seminars & Events for Ergodic Theory & Statistical Mechanics

##### Attractors with Large Invisible Parts

Philosophically, an attractor of a dynamical system is a closed subset of the phase space which orbits "approach" as time goes to infinity. Different meanings of the word "approach" produce different versions of attractors: maximal, Milnor, statistical, minimal etc. The question of generic non-coincidence between these various types of attractors has not yet been answered. We will be concerned with a different point of view. Physically, if one looks at the attractor, then one knows where most orbits will go to as time goes to infinity. But it is possible that a large part of the attractor is redundant, in the sense that orbits spend very very little time near it. Thus, it would be more significant to look only at the non-redundant part of the attractor.

##### Quenched Central Limit Theorem for Random Toral Automorphism

The statistical properties of the Lorentz gas with periodically positioned obstacles are well understood. The random case, obtained after each of the obstacles undergoes a small i.i.d. displacement, stands as a challenge. The latter can be studied in terms of a random sequence of hyperbolic symplectic (billiard) maps, which however is not i.i.d. due to recollisions. In fact, even the i.i.d. sequence (no recollisions) is poorly understood.

Motivated by the above, we study an i.i.d. sequence of toral automorphisms in two dimensions. We will argue that the time-$N$ average of any observable has Gaussian fluctuations of order $\sqrt{N}$ for almost every sequence of maps, and that the variance is independent of the sequence. Joint work with Arvind Ayyer (Rutgers) and Carlangelo Liverani (Rome 1).

##### Exits from an infinite tube

We consider a billiard system in an infinite tube with periodic scatterers. We show that with probability 1 a particle exits from the tube. Surprisingly, the probability that the exit velocity is opposite to the initial one tends to 1 in a limit when the size of scatterers vanishes.

##### The bilinear Hardy-Littlewood function for the tail

##### Spherical billiards with many 3-periodic orbits

It is known that the Lebesgue measure of 3-periodic trajectories in a planar (Birkhoff) billiards is zero (and a well-known conjecture states that the same is true for any period). On the sphere, however, it is easy to construct a billiard domain with 2-dimensional family of 3-periodic orbits (take the intersection of the sphere with the positive octant). In this talk I will explain why this is essentially the only possible construction.

##### Logarithm laws for horocycles

In joint work with G. Margulis, we prove a logarithm law for unipotent flows on the space of unimodular lattices in $R^n$.

##### Trigonometric sums and continued fractions with even partial quotients

I will talk about the geometric features ("curlicues") of quadratic trigonometric sums and discuss how the renormalization of such sums is connected with continued fraction expansions with even partial quotients. I will also explain a recent renewal-time limit theorem for the sequence of denominators generated by such expansions.

##### Invariant curves near the boundary of an annulus without a twist hypothesis, following M. R. Herman

Sometime in the nineties, M. R. Herman gave a series of lectures at Columbia on KAM theory. Yasha asked me to speak on one of the results that Herman discussed in his lectures. Here is the result:

Let $f$ be an area preserving infinitely differentiable diffeomorphism of a closed annulus. Suppose that the restriction of $f$ to one of the boundary components is a rotation whose rotation number satisfies a Diophantine condition. Then there exist an infinite number of rotational invariant curves in an arbitrarily small neighborhood of the given boundary component.

##### On the quantitative equidistribution of nilfows and Weyl sums

It is know since the work of Furstenberg that the equidistribution of the fractional parts of polynomials sequences with irrational leading coeeficient can be derived from the unique ergodicity of (certain) nilflows. We will present some results on the speed of convergence of ergodic averages of nilflows under Diophantine conditions and discuss the relation with known results and conjectures on bounds of Weyl sums (exponential sums for polynomial sequences). The method of proof is based on the analysis of the action of a suitable 'renormalization' on the space invariant distributions for nilflows (in particular it makes no use of number theory). The content of this talk is joint work with L. Flaminio.

##### Nonconvergence examples in averaging

Systems which combine fast and slow motions lead to complicated two scale equations and the averaging principle suggests to approximate the slow motion by averaging in fast variables. When the fast motion does not depend on the slow one this approximation usually works for all or almost all initial conditions but when the slow and fast motions depend on each other (fully coupled), as is usually the case, the averaging prescription cannot always be applied, and when it is valid then only in the sense of convergence in measure (or in average) with respect to initial conditions. A nonconvergence example for fixed initial conditions constructed for small perturbations of integrable Hamiltonian fast motions is due to Neishtadt and it is based on the well known phenomenon of resonances there.