# Seminars & Events for Ergodic Theory & Statistical Mechanics

##### The decay of Fourier modes in solutions of Navier-Stokes systems

##### Nearest neighbor distances for several rotations

We will discuss results of Marklof on distributions of nearest neighbor distances. We will look at the Poisson scaling and at CLT scaling. Another point of view is to look at the number of distinct gap lengths in this scenario. Here we will explain unpublished results of Boshernitzan and Dyson.

##### A priori bounds for bounded-primitive renormalization

We say that an infinitely renormalizable quadratic polynomial has bounded-primitive type if we can find an infinite sequence of primitive renormalization times, such that the ratio between consecutive terms of the sequence is bounded. We prove that any such polynomial has the a priori bounds: there is a lower bound on the modulus of all renormalizations. This implies that the Mandelbrot set is locally connected at the associated parameter values.

##### A pair correlation bound implies the Central Limit Theorem for Sinai Billiards

It is an open problem in the study of dynamical systems whether fast decay of correlations alone is sufficient for the Central Limit Theorem (CLT) to hold. On the one hand, there are no examples of dynamical systems for which correlations decay quickly but the CLT fails. On the other, existing CLT proofs rely on statistical properties much stronger than correlation decay. In the talk I will discuss a prime class of physically relevant systems, called Sinai Billiards, and show that a single bound on correlations indeed implies the CLT directly. As a byproduct, the CLT is obtained for observables possessing remarkably little regularity.

##### Ergodicity of some boundary driven integrable Hamiltonian chains

Small Hamiltonian systems are connected in a chain the ends of which are coupled to unequal heat baths, forcing the system out of equilibrium. Energy exchange is of a form that leads to integrable dynamics. A proof of ergodicity of both equilibrium and nonequilibrium steady states will be presented. This is followed by numerical results which show that unlike certain mechanical systems with chaotic microdynamics, marginal distributions of NESS in these integrable chains are not Gibbsian, leading to problems in the definition of “local temperature.”

##### A brief survey of effective equidistribution results in Gamma\G

Equidistribution results for orbits and more general configurations in Gamma\G are a central focus of the theory of flows on homogeneous spaces. A notable example that comes to mind is Ratner's equidistribution theorem. I will survey some old and new quantitive equidistribution results of this flavor by several authors.

##### Unbiased Random Perturbations of Navier-Stokes Equation

A random perturbation of a deterministic Navier-Stokes equation is considered in the form of an Stochastic PDE with Wick product in the nonlinear term. The equation is solved in the space of generalized stochastic processes using the Cameron-Martin version of the Wiener chaos expansion. The generalized solution is obtained as an inverse of solutions to corresponding quantized equations.

An interesting feature of this type of perturbation is that it preserves the mean dynamics: the expectation of the solution of the perturbed equation solves the underlying deterministic Navier-Stokes equation. From the standpoint of a statistician it means that the perturbed model is unbiased. The talk is based on a joint work with R. Mikulevicius.

##### Diophantine Properties of Dynamical Systems and IETs

The lecture is based on a recent preprint with the same title, joint with J. Chaika and put recently on arXiv. One of the results is that for ergodic IETs (Interval Exchange Transformations) almost sure $\liminf_{n\to\infty} n|T^nx-y|=0$.

The result is optimal in two ways: \\ (1) the normalizing factor \ $n$ \ cannot be improved, even for rotations;\\ (2) the assumption of ergodicity cannot be replaced by just minimality.

##### The spectral dichotomy for one-frequency Schrödinger operators

In the theory of one-frequency Schrödinger operators, the best understood potentials have been those that can be somehow considered either small or large. Roughly, small potentials tend to inherit the behavior of the Laplacian and present absolutely continuous spectral measures (leading to good transport properties), while for large potentials it is Anderson localization that prevails.

##### Convergence of renormalizations

##### Nodal sets for eigenfunctions of the Laplacian and lattice points on circles and spheres

##### Limit theorems for sticky particle systems and positivity of integrated random walks

Consider the model of a one-dimensional gas, whose particles have random initial positions and random initial velocities. Particle attract each other due to gravitation, and stick together at collisions. As time goes, the number of particles decreases while their sizes increase until there forms a giant single particle of the total mass.

##### On the limit curlicue process for theta sums

I shall discuss a random process achieved as the limit for the ensemble of curves generated by interpolating the values of theta sums. The existence and the properties of this process are established by means of purely dynamical tools and rely on generalizations of a result by Marklof and Jurkat and van Horne. (joint work with Jens Marklof).

##### The dimension of self-affine sets: past, present and future.

Calculating the dimension of sets invariant under non-conformal dynamics is a formidable problem. My talk will be a survey on what is known and expected for self-affine sets, i.e. sets invariant under piece-wise affine expanding maps on Euclidean space. Some emphasis will be given to my joint work with A. Käenmäki on self-affine sets of Kakeya type, and the thermodynamic formalism for the singular value function.

##### Random polygons in plane convex sets

Consider picking $N$ random points in a convex set $K$ and forming their convex hull $K_N$. Recently, there have been a number of results concerning the asymptotic behavior of random variables such as the area and number of vertices of $K_N$. These are, however, all limited to two special cases: 1) $K$ is a polygon and 2) $K$ is "smooth". I will discuss work which obtains uniform bounds over the family of all convex sets $K4$. These results include central limit theorems for the area and number of vertices of

$K_N$.

##### The Oseledets basis for products of non-identically distributed independent random matrices

The famous Oseledets theorem states that if $g_n$ is a stationary sequence of $m\times m$ matrices then with probability 1 there is a (random) basis in ${\mathbb{R}}^m$ such that for any vector $x$ the asymptotic behaviour of $||g_n...g_1x||$ is the same as that for one of the vectors from this basis. The fact that the sequence is stationary is crucial for the existence of such a basis. I shall consider the case when the sequence is not stationary. It turns out that for independent non-identically distributed matrices one can still obtain an analogue of the Oseledets's results. Some applications will be considered.

##### Independence properties of weakly mixing systems and polynomial pattersn in large sets

Various recurrence and convergence results obtained in recent years indicate that dynamical systems exhibit regular behavior along polynomial times. In particular, weakly mixing systems turn out to always possess rather strong independence properties along certain sets of zero density. We will discuss some implications of these results in physics as well as applications to combinatorics and number theory (including polynomial extensions of Szemeredi's theorem on arithmetic progressions and recent work of Tao and Ziegler on polynomial patterns in primes). We will also formulate and discuss some natural open problems and conjectures.

##### Concentration inequalities for dynamical systems

Concentration inequalities are a powerful tool to estimate the fluctuations of observables more general than ergodic sums: one can consider any observable $F(x,\ldots,T^n x)$ provided it is separately Lipschitz. Such inequalities can be established for non-uniformly hyperbolic systems and we shall present some applications.

##### Applications of Expansion and Equidistribution to Number Theory

We will discuss recent applications of Expander Graphs to various problems in Equidistribution and Sieving.

##### Dynamics of bouncing balls

We consider a ball bouncing off infinitely heavy periodically moving wall in the presence of a potential force. We are interested in the question how large is the set of orbits whose energy tends to infinity. Both smooth and piecewise smooth motions of wall will be considered. We also present some related questions about small piecewise smooth perturbations of nearly integrable systems.