Spring 2006 Workshop on Dynamical Systems and Related Topics
in honor of Yakov G. Sinai on the occasion of his 70th birthday

Abstracts

Michael Aizenman, Princeton University
Persistence under weak disorder of AC spectra of quasi-periodic Schroedinger operators on trees graphs

An outstanding challenge in the theory of random operators is to explain the conditions under which absolutely continuous spectra will persist in the presence of disorder. The talk will focus on this question in the context of the discrete Schroedinger operators on tree graphs, with the potential given by the combination of a quasi-periodic radial term and a random stationary perturbation. We show that while the AC spectrum disappears under any radially symmetric disorder, it is stable if the disorder is weakly correlated, e.g., independent, provided the following condition is met. The sufficiency criterion is the existence of Bloch-Floquet states for the one dimensional operator corresponding to the radial problem. The existence of Bloch-Floquet states for quasi-periodic operators is a topic which has been strongly developed since the early works by Dinaburg and Sinai (1975). The condition is related to the reducibility of the Schroedinger cocycle (Eliason 92, Puig 05). Our analysis proceeds through the study of the fluctuations of the random operator's Green function, and makes an essential use of the cocycle reducibility, where available. (Joint works with S. Warzel, and R. Sims.)

Pavel Batchourine, Princeton University
Singularity manifolds and fundamental theorem for multidimensional dispersing billiards

Stochastic properties of dispersing billiards have been studied since 70's, but recently it was discovered that additional information about so-called singularity manifolds is needed in order to prove, for exmaple, ergodicity. I shall present the results on the structure of these manifods and explain how they help to recover the proof of ergodicity of dispersing billiards.

Pavel Bleher, IUPUI
Critical behavior of Gaussian random matrices with external source

Abstract in PDF format

Leonid Bunimovich, Georgia Tech
Continued fractions and billiards

In his seminal paper on billiards (Russian Mathematical Surveys, 1970) Sinai developed a general machinery for proving ergodicity and K-property for nonuniformly hyperbolic dynamical systems. Particularly he introduced certain continued fractions that correspond to the orbits of biliards and define the local stable and unstable fibers for an orbit. Moreover, there is a simple relation between the Kolmogorov-Sinai entropy of hyperbolic billiards and these continued fractions. This relation is widely used in the modern nonequilibrium statistical mechanics. It turned out that to any continued fraction with real valued elements can be corresponded an orbit of some "virtual" billiard. This correspondence allows to obtain some new theorems on convergence of continued fractions.

Alexander Bufetov, University of Chicago
The central limit theorem for the Teichmueller flow on the moduli space of Abelian differentials

The talk will be devoted to the Central Limit Theorem for the Teichmueller flow on the moduli space of abelian differentials with a prescribed pattern of singularities. The proof follows the scheme introduced by Sinai for geodesic flows on manifolds of negative curvature.

The first step is a representation of the Teichmueller flow as a suspension flow over the natural extension of the Rauzy-Veech-Zorich induction map on the space of interval exchange transformations. In genus one, this construction corresponds to a representation of the geodesic flow on the modular surface as a suspension flow over the natural extension of the Gauss continued fraction map.

The main step of the proof is a stretched-exponential bound on the decay of correlations for the Rauzy-Veech-Zorich induction map. The induction map admits a natural symbolic representation over a countable alphabet, and the decay of correlations is obtained by the method of Markov approximations of Sinai, Bunimovich-Sinai. After that, the Theorem of Melbourne and Torok completes the proof.

Nikolai Chernov, University of Alabama
Slow decay of correlations in a dispersing billiard table with cusps (a joint work with R. Markarian)

Consider a billiard table made up by three circlular arcs (of the same radius) that are tangent to each other at their endpoints; it is a dispersing table with three cusps. It was predicted in 1983 by J. Machta (based on a heuristic argument) that the velocity autocorrelation function should decay as O(1/n), where n is the counter of collisions. We study this model rigorously, using Young's tower construction, to estimate the rate of decay of correlations for Holder continuous functions. This model exhibits a surprisingly complicated behavior when trajectories are trapped in cusps for a long time. We establish a nearly optimal bound on correlations. This result is relevant to the studies of Brownian motion of a heavy disk submerged in an ideal gas of light particles in a box.

Weinan E, Princeton University
The crystallization problem

Why is the ground state of a solid a crystal lattice? In this talk, we will review recent attempts on establishing this result rigorously, starting from atomistic models of solids.

Charles Fefferman, Princeton University
Whitney's extension problems

How to decide whether a given subset of R^n is contained in a C^m smooth hypersurface. Also, joint work with Bo'az Klartag on a version for large finite sets.

Susan Friedlander, University of Illinois-Chicago
The limit of vanishing viscosity for the Navier-Stokes spectrum

We show that the eigenvalues of the Navier-Stokes operator, in the limit of vanishing viscosity, converge precisely to those of the underlying Euler operator beyond the essential spectrum. This is joint work with Roman Shvydkoy.

Michael Goldstein, University of Toronto
Resonances and formation of the gaps in the spectrum of quasi-periodic Schrödinger equation

Abstract in PDF format

Dmitry Jakobson, McGill University
Estimates from below for the remainder in local Weyl's law

We obtain asymptotic lower bounds for the spectral function of the Laplacian and for the remainder in local Weyl's law on compact manifolds. In the negatively curved case, thermodynamic formalism is applied to improve the estimates. Our results can be considered pointwise versions (on a general manifold) of Hardy's lower bounds for the error term in the Gauss circle problem.

Svetlana Jitomirskaya, UC Irvine
Quasiperiodic operators with analytic potential at low coupling: sharp results

We will discuss recent results on analytic Schrodinger cocycles at small couplings, with applications including the dry version of Ten Martini problem and 1/2-Holder continuity of the integrated density of states for Diophantine frequencies. The estimates on the coupling are independent of frequency and are optimal in certain cases. Bloch structure of solutions is equivalent to the analytic reducibility of cocycle and is linked to the existence of localized eigenfunctions for the dual model. However, there is a generic set of energies in the spectrum for which no localized eigenfunctions exist. For such energies the solutions are still linked through a possibly divergent Fourier series. We show that for all energies there are solutions (for the dual model) that are localized on a large set, between a sparse sequence of resonances. This allows to give sharp estimates on the dynamics (and therefore solutions) for all energies. This is joint work with Artur Avila.

Anatole Katok, Pennsylvania State University
Rigidity for higher rank actions via KAM: partially hyperbolic, elliptic, and possibly parabolic cases

Moser's seminal 1990 paper on commuting circle diffeomorphisms introduced a version of KAM scheme where the system of conjugacy equations is considered as over-determined. Moser used this scheme to overcome problems appearing when individual diffeomorphisms have abnormally well approximable rotation numbers and satisfy Diophantine conditions only jointly. While recent developments in the circle case deal with the global situation and hence do not use KAM, Moser's idea turned out to be very fruitful for the treatment of actions with completely different types of dynamics, where genuine rigidity appears: actions by commuting partially hyperbolic automorphisms of the torus (D. Damjanovic--A.K., 2004) and nil-manifolds (Damjanovic, in progress) and, most surprisingly, possibly even some unipotent actions such as the diagonal action on the Cartesian square of SL(2,R) factored by an irreducible lattice. Possibility of using the KAM scheme in the latter and similar unipotent cases is based on the study of the cohomological equation in the higher--rank parabolic case in the Ph. D. thesis by D. Mieczkowski (PSU, 2006). His method uses some estimates obtained by L. Flaminio and G. Forni in their 2002 paper for the rank one case as well as a version of the ``higher--rank trick'' to establish vanishing of obstructions.

In this talk I will give a brief overview of this version of KAM method and discuss the key elements in its applications to various cases.

Yuri Kifer, Hebrew University
Probabilistic problems in deterministic fully coupled averaging

The averaging setup arises in the study of perturbations of dynamical systems with constants of motion which give rise to a combination of fast and slow motions. Such problems emerged initially in celestial mechanics and they lead to complicated multiscale equations. Classical averaging setup deals mainly with perturbations of integrable Hamiltonian systems but considering perturbations of families of hyperbolic and expanding dynamical systems, among them some of nonintegrable Hamiltonian systems, we arrive at stochastic behaviour of the slow motion which cannot be observed in the classical framework. Among motivations for such study are some models of climate-weather interactions where climate is viewed as a slow and weather as a fast chaotic motion. We discuss mainly large deviations of the slow motion from the averaged one which lead to a probabilistic type description of its very long time (adiabatic) behaviour such as exits from neighborhoods of attractors of the averaged motion and rare transitions between them. Even perturbations of simple families of expanding maps of the interval where computer simulations are easy to perform yield interesting nontrivial problems some of them still unanswered.

Leonid Koralov, Princeton University and University of Maryland
Averaging of Hamiltonian flows with an ergodic component

We consider a process which consists of the fast motion along the stream lines of an incompressible periodic vector field perturbed by the white noise. Together with D. Dolgopyat we recently showed that for almost all rotation numbers of the unperturbed flow, the perturbed flow converges to an effective, "averaged" Markov process. This is a generalization of classical results of Freidlin and Wentzell, who considered the case when all the flow lines of the unperturbed flow are closed curves. We shall also discuss some related problems on the long time behavior of randomly perturbed periodic flows.

Gregory Margulis, Yale University
Closed orbits of group actions

Counting problems and asymptotic behavior of closed orbits of actions of different kinds of groups will be discussed. I will start with the classical case of one-parameter group and after that talk about the cases of semismple and multidimensional groups.

John Mather, Princeton University
Minimization and averaging

In my proof of Arnold Diffusion in two and a half and in three degrees of freedom, it is important to understand the relationship between c-minimal orbits of the given system (a small perturbation of an integrable system) c-minimal orbits of various averaged systems. In this talk, I will describe useful results of this type.

Jonathan Mattingly, Duke University
The stochastic Navier Stokes equation: ergodicity and spectral gaps

I will discuss recent progress in understanding mixing and ergodicity in stochastically forced PDEs. I will use the 2D stochastic Navier Stokes equation as my primary example. I will concentrate on the case where the forcing is degenerate in that all of the degrees of freedom are not directly agitated stochastically. This hyopelliptic setting will require an infinite dimensional version of Hormander's "sum of squares theorem". The discussion will include the use of ideas from Malliavin calculus and a replacement for the classical strong Feller property for Markov semi-group.

David McClendon, University of Maryland
Orbit discontinuities of Borel semiflows on Polish spaces

Let $X$ be a standard Polish space. Given an action $T_t$ of $[0,\infty)$ by (presumably non-invertible) Borel maps on $X$, we say that two distinct points $x$ and $y$ are ``instantaneously discontinuously identified’’ (IDI) if $T_t(x) = T_t(y)$ for all $t > 0$. Such phenomena is of interest because it is the obstacle to representing the action as a shift map on a space of continuous paths. We define the concept of ``orbit discontinuity’’, a generalization of IDI, and discuss results regarding the structure and prevalence of such behavior. In particular, given any $x \in X$, the set of times $t$ for which $T_t(x)$ is IDI is a countable set.

Sheldon Newhouse, Michigan State University
Homoclinic phenomena on surfaces

We will survey some recent developments in connection with homoclinic tangencies on surfaces. In particular, we describe consequences for the existence of so-called Sinai-Ruelle-Bowen measures, Hausdorff dimension, and the existence of symbolic extensions.

Valery Oseledets, Moscow State University
Erdos measures and Markov chains

Abstract in PDF format

Natasa Pavlovic, Princeton University
On periodic nonlinear Schrödinger equations

In this talk we will present a joint work with Daniela De Silva, Gigliola Staffilani and Nikolaos Tzirakis on global well-posedness for the $L^2$ critical Schr\"{o}dinger equation with periodic boundary conditions in 1D and 2D. By combining an implementation of the method of almost conservation laws with number theoretic techniques we prove that the problem is globally well-posed in 1D in the Sobolev space $H^{s}({\Bbb T})$, for any $s>4/9$ and in 2D in the Sobolev space $H^{s}({\Bbb T}^2)$, for any $s>2/3$.

Our 1D result matches the best known global well-posedness result for the corresponding problem on line. The two dimensional result was already announced by Bourgain while discussing the possible exponent $s$ that the method of almost conservation laws would give in this context. While explicitly writing up the calculations to recover this claim, we noticed that in one particular case, a better Strichartz inequality was needed to successfully conclude the argument. We proceed by determining a qualitative $\epsilon$ refined Strichartz type estimate which reduces to counting the lattice points on a "small" portion between two concentric circles.

David Ruelle, IHES
Nonequilibrium statistical mechanics of a classical infinite system of rotators

We study the dynamics of an infinite system of coupled classical rotators. An initial state with infinite thermostats evolves into "smooth states", for which the entropy of finite regions is defined. We propose a definition for the local rate of entropy production, and show that some expected properties of this quantity are satisfied.

Michael Shub, University of Toronto
Entropy estimates for circle mappings

We discuss two families of immersions of the circle which are formed by fixing one immersion and composing with rotations. One family starts with a Blaschke product the other with the map x ---> kx + \epsilon sin(2 \pi x). In the first case we give a lower bound estimate for the average entropy in the family. In the second case we show how this estimate mildly fails as a function of \epsilon. This is joint work with Leonel Robert and Enrique Pujals in the first case and Carles Simo and Rafael de la LLave in the second.

Corinna Ulcigrai, Princeton University
Mixing for suspension flows over interval exchange transformations

We consider special flows over interval exchange transformations with an asymmetric logarithmic singularity of the roof function. We prove that that for a full measure set of IETs the suspension flow is strongly mixing. This generalizes a result by Sinai and Khanin for special flows over a typical rotation of the circle. We will also explain the connection ofthese type of suspension flows with hamiltonian flows on surfaces of genus g.

Paul Wright, New York University
A simple piston problem

A simple model of a piston consists of a heavy point particle of mass M moving inside the unit interval. On either side there are a finite number of light gas particles which do not interact with each other, but which interact with the walls at 0 and 1 and the heavy particle via elastic collisions. The problem is to find the limiting dynamics of the piston on a certain time interval as the mass M tends to infinity. I learned about this problem from the papers of Neishtadt and Sinai, who pointed out that a classical averaging theorem due to Anosov could be extended to this case and derived the averaged equations. I will discuss a new proof which strengthens these results, as well as various generalizations, including what happens when the particles interact via a soft potential.

James Yorke, University of Maryland
The edge of chaos in a fluid flow

How does one describe high dimensional chaotic systems? I will report on numerical studies of a 9 dimensional ordinary differential equation representing a fluid flow between two boundaries. The system has a fixed point attractor and transient chaos, that is, some kind of generalized horseshoe with an invariant set having dimension approximately 5. I will describe our findings. This is joint work with Joe Skufca and Bruno Eckhardt.