Lorentz Center - Nonlinear Dynamics, Ergodic Theory and Renormalization from 20 Sep 2004 through 24 Sep 2004
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### Nonlinear Dynamics, Ergodic Theory and Renormalizationfrom 20 Sep 2004 through 24 Sep 2004

No Title

A. Avila

Weak mixing for interval exchange transformations and translation flows (with G. Forni)

Abstract: We prove weak mixing for almost every interval exchange transformation which is not a rotation. We also get weak mixing for almost every translation flow on surfaces of higher genus. The proof involves a statistical analysis of the dynamics of the renormalization operator.

Y. Avron

The Integer Quantum Hall Effect from a geometric point of view.

Abstract: I shall review theories of the integer quantum Hall effect where the Hall conductance is identified with a Chern number, in one setting, and with a Fredholm index in another.

H. van Beijeren

Collective Lyapunov modes in hard disk systems (with Astrid de Wijn)

Abstract: Computer simulations by Posch and coworkers show that the smallest positive Lyapunov exponents of a system of many moving hard disks are characterized by collective modes of sinusoidal structure. The Goldstone mechanism relates these modes to global symmetries of the system. A generalized Boltzmann equation may be used to calculate the Lyapunov exponents under consideration. However, in the limit of vanishing density small discrepancies remain with the values obtained by the computer simulations. These may be attributed to ring corrections to the Boltzmann equation.

M. Benedicks

Parameter selection and positive Lyapunov exponent for non-uniformly hyperbolic dynamical systems

In the talk I will review how the techniques of selecting parameters in low dimensional dynamical systems to prove positive Lyapunov exponent have evolved during the last 10-15 years. The first examples are quadratic maps and maps from the H\'enon family where positive Lyapunov exponents have been proved by Benedicks and Carleson for a set of positive Lebesgue measure in the parameter space. Another examples are {\it Viana maps}, which are skew product, where quadratic maps are driven by expanding maps. Here improvements of the original results of Viana have been obtained by Buzzi, Sester and Tsujii and most recently in a masters thesis of Daniel Schnellmann at KTH. Yet another example is positive Lyapunov exponent for a positive measure set in the parameter space of rational maps. This is proved in Magnus Aspenberg's recent thesis at the KTH extending a famous theorem of Mary Rees, where existence of an absolutely continuous invariant measure is proved in this setting.

J. van den Berg

Self-organized critical forest fire models in one and two dimensions

Consider the forest-fire model where trees grow with a fixed rate (say 1) on the vertices of a lattice, and where trees are hit by lightning at rate lambda. When a location is hit by lightning, its entire cluster of trees disappears instantaneously.

Such models have been studied extensively in the literature, especially by physicists. The most interesting behaviour seems to occur when the lightning rate tends to 0. It is believed that then the system shows so-called self-organized criticality. For the two-dimensional case there are very few rigorous results, and even from a heuristic point of view the situation is far from clear. I will briefly present a percolation-like conjecture and its relations to some essential 2D forest-fire problems. Then I will discuss the one-dimensional forest-fire model, which (in the physics literature) was thought to be completely understood. However, as I will show, some of the results need significant correction.

This talk is based on joint work with Rachel Brouwer and joint work with Antal Jarai.

J. Bricmont

Fourier's law and the approach to equilibrium for coupled anharmonic oscillators

Abstract: We consider a system of coupled anharmonic oscillators on a d-dimensional lattice, with noise and friction acting on each site. The Gibbs states are the stationary states for this stochatsic process and perturbations of those states vanish exponentially fast under the stochastic dynamics. One would like to show that a small perturbation of the stationary states has a diffusive evolution, in the limit where the noise and the friction go to zero. As a step in that direction, one shows that a truncated system of equations for the two point function of that system behaves diffusively in an appropriate scaling limit.

H.P. Bruin

Existence of invariant probabilities for interval maps without growth conditions on derivatives

The existence of an absolutely continuous invariant probability (acip) for an interval maps with a critical point has been tied to the growth of derivatives along the orbit of this critical point. Whereas the first papers in this direction showed that exponentential growth (Collet \& Eckmann, Nowicki) or stretched exponential growth (Jakobson) were sufficient, the summability condition of Nowicki \& van Strien has been the strongest existence result for over a decade. This talk will show that no growth to infinity is needed. An acip can be shown to exist provided $\liminf |Df^n(fc)|$ is sufficiently large. The result also shows that for e.g. the quadratic family $f_a,$ maps without an acip can only be found close (in some sense) to parameter regions where $f_a$ is renormalizable

T. Dorlas

Quantum coding

This is a review of quantum coding concentrating on noiseless coding theory. We start with a brief overview of Shannon's classical coding theory. Then we show how this can be generalised to a quantum setting and give a short proof of the Hiai-Petz theorem which we will argue is a crucial part of quantum coding theory.

F. Dumortier

Cocoon bifurcation in three-dimensional reversible vector fields

Abstract: The cocoon bifurcation is a set of rich bifurcation phenomena numerically observed in the three-dimensional ODE system for travelling waves of the Kuramoto-Sivashinsky equation. In this talk we present the "cusp-transverse heteroclinic chain" as organizing center of a basic part of the cocoon bifurcation in the more general setting of reversible vector fields on R3. We also discuss the relation with a heteroclinic cycle called the reversible Bykov cycle. The talk is based on results obtained in collaboration with S. Ibanez and H. Kokubu.

L.H. Eliasson

kam-theory for the non-linear Schrödinger equation

G. Gentile

Degenerate lower-dimensional tori and resummations of divergent series

Abstract: Quasi-periodic motions on invariant tori of an integrable system of dimension strictly smaller than half the phase space dimension (lower-dimensional tori) may continue to exist under small perturbations. Such tori, in absence of perturbations, are degenerate, in the sense that the normal frequencies are vanishing. I shall consider a class of systems for which the parametric equations of the invariant tori can be computed as formal power series in the perturbation parameter (Lindstedt series): a resummation algorithm for the series can be devised and proved to be convergent provided that the perturbation parameter is small enough, so implying existence of both hyperbolic and elliptic lower-dimensional tori for the perturbed systems. In the latter case the perturbation parameter has to fulfil the further condition that it has to belong to a suitable Cantor set of large relative Lebesgue measure, which is characterised by imposing infinitely many Diophantine condition of Melnikov type.

W.Th.F. den Hollander

Renormalization of interacting diffusions

Abstract: Systems of hierarchically interacting diffusions allow for a rigorous renormalization study. In this talk we consider systems of coupled SDE's of the type

 dXi(t) = å j Î WN aN(i,j)[Xj(t)-Xi(t)] dt+ Ö g(Xi(t)) dWi(t),

 i Î WN, t ³ 0, Xi(t) Î S,

where S is the single-component state space, WN is the hierarchical lattice of order N, aN(·,·) is an (appropriately chosen) interaction kernel on WN×WN, g(·) is the single-component diffusion function on S, while {Wi(·)}i Î WN is a collection of independent (standard) Brownian motions that drives the evolution. As initial condition we take {Xi(0)}i Î WN to be a stationary and ergodic random field with mean q Î int(S).

We define block averages on space-time scale k by putting

 Yi,N[k](t) = 1 Nk å [(j Î WN) || (||j-i|| £ k)] Xj(Nkt),       i Î W, t ³ 0, k Î \mathbb N.

It is expected that

 Y[k](t) = lim N®¥ Y0,N[k](t)

evolves according to an autonomous SDE of the type

 dY[k](t) = [q-Y[k](t)] dt+ Ö (Fkg)(Y[k](t)) dW(t)

with initial value Y[k](0)=q, where Fkg is the diffusion function on scale k, obtained from the diffusion function g on scale 0 by applying k times a renormalization transformation F.

We describe several cases where the above renormalization scenario has been rigorously established, namely,

 S=[0,1], [0,¥), [0,¥)2,     g Î H(S),

with H(S) the class of Lipschitz functions on S that are strictly positive on int(S) and vanish appropriately on S. It turns out that F is a non-linear integral operator with a rich structure of fixed points and attracting orbits. The latter correspond to special choices of g, for which the system has certain nice duality properties. They play the role of universal attractors for the dynamics on the macroscopic scale (corresponding to k®¥).

W. Kager

Stochastic Löwner Evolutions: scaling limits of critical models

Abstract: Stochastic (or Schramm-) Löwner Evolution has appeared as a new tool for understanding and studying the scaling limits of critical models. Invented by Schramm, the subject of SLE was further developed mainly in a collaboration of Lawler, Schramm, and Werner. The purpose of this lecture is threefold:

1.      to explain what SLE is,

2.      to explain how the connection with critical models is made, and

3.      to give examples of how critical exponents can be computed using SLE.

R. Krikorian

Renormalization of quasi-periodic cocycles and applications

Abstract: We describe renormalization procedures for $SU(2)$ valued and $SL(2,{\bf R})$-valued quasi-periodic cocyles and show how to use them to address the reducibility problem for such cocycles. We show for example that reducible $SU(2)$-valued cocyles (with fixed typical frequency) are $C^\infty$ dense. On the other hand (this is a joint work with Artur Avila) almost all $SL(2,{\bf R})$-valued cocycles are either smoothly reducible or have positive Lyapunov exponent. We also give some application of this last result to the spectral theory of the qp Schrodinger operator (Aubry-Andre conjecture).

J.S.W. Lamb

The geometry of Penrose tilings: projection and renormalization

Abstract: In the early 1970's, R. Penrose constructed a set of two tiles that can tile the plane only nonperiodically. His proof uses a renormalization argument based on the existence of substitution rules. In 1981, N. de Bruijn showed that Penrose tiling also can be obtained by projection of a discrete plane in R5 (with vertices in Z5) to the nearest two-dimensional hyperplane. In this talk we show that projection tilings of this type admit (a countable infinity of different) substitution rules if and only if there exists a "quadratic" hyperbolic lattice automorphism that fixes the projection hyperspace. As the latter condition is very easy to verify, we obtain a simple characterization (and many new examples) of such renormalizable projection tilings. This is joint work with Edmund Harriss.

M. Lyubich

tba

H.A. Posch

Lyapunov modes and other peculiarities of the tangent-space dynamics of fluids

Abstract: The perturbations associated with the small Lyapunov exponents of a fluid exhibit coherent stationary patterns in physical space, to which we refer as "Lyapunov modes". They were first observed for hard-disk fluids in one, two and three dimensions. Using refined Fourier-transformation methods, we demonstrate that they exist also in soft-disk systems. We discuss the symmetry properties and the dynamics of the modes.

In addition, we study also many-particle systems in stationary nonequilibrium states and review some recent results on the phase-space fractals and their projections onto subspaces associated with non-thermostatted degrees of freedom. Both dynamical and stochastic thermostats are considered.

J. Pöschel

On the well-posedness of the periodic KdV equation

Abstract: We describe results about the well-posedness of the periodic KdV equation in weighted Sobolev spaces, which include for example Gevrey-type spaces. The argument is based on the existence of global Birkhoff coordinates for KdV.

J. Puig

Reducibility of quasi-periodic skew-products and the spectrum of Schrödinger operators

Abstract: In this talk we will see how spectral properties of Schrödinger operators with quasi-periodic potential can be derived through an analysis of the dynamics of the corresponding eigenvalue equations and the skew-products they define. A central tool will be the concept of reducibility of quasi-periodic skew-products to constant coefficients. As an application to Schrödinger operators, Cantor spectrum will be obtained in some models and, in particular, in the Almost Mathieu or Harper case.

E. Pujals

Contribution to the proof of the conjecture about density of heteroclinic cycles, homoclinic tangencies and hyperbolicity.

Abstract: We show the proof of the following theorem: Let f be a Kupka-Smale C2 diffeomorphisms acting on a three dimensional manifold. Let Hp be an attracting homoclinic class associated to a periodic point (Hp=Çn > 0 fn(U) for some neighborhood U of Hp). Then, one of the following statements holds:

- Hp is a hyperbolic homoclinic class;
- f can be C1 approximated by a diffeo g exhibiting a homoclinic tangency;
- f can be C1 approximated by a diffeo g exhibiting a heteroclinic cycle.

We will also show how this result can be extended to higher dimensions.

R. Roussarie

Limit cycles in Liénard equations

Abstract: Many partial results are known regarding the classical Liénard equations

 × x

 =

 y - Fa (x),

 × y

 =

 -x.

Here Fa is a polynomial of degree 2k+1, Fa (x) = åi=22k+1 ai xi where a = (a2, ¼,a2k+1) Î R2k.

For instance, it can be directly seen that for any a the related vector field Xa has only a finite number of limit cycles. Indeed, this comes from the fact that Xa has a return map, globally defined on the half axis Ox = {x ³ 0}, and that this map is analytic. Also it is easy to verify that at most k limit cycles can bifurcate from the origin. For these two reasons, De Melo, Lins and Pugh have conjectured that the total number of limit cycles, on the whole (x,y)-plane and for all a, also is bounded by k. However, it is not even known whether there exists a finite bound L(k) independent of a, for the total number of limit cycles.

In this talk the above problem will be addressed, showing that it is related to problems concerning the finiteness of the number of limit cycles which bifurcate from Canard cycles of singular differential equations, obtained as singular limits of the Liénard equations for large values of |x| + |y| or |a|. Also generalized Liénard equations will be considered in the talk.

E. Verbitsky

Transformations of Gibbs measures

Abstract: Transformations of Gibbs measures appear naturally in dynamical systems, statistical mechanics and information theory. The two most frequent question are:

Is the transformed measure Gibbs?
What is the entropy of the transformed measure?

In this talk I will discuss two examples: the fuzzy Gibbs measures, which are obtained by a uniform coding of the phase space, and the jittered measures, which serve as models of error propogation in optical storage (CD's, DVD's, etc). For the fuzzy Gibbs measures I present a criterium for Gibbsianity. For the jittered measures I will discuss two methods of computing the entropy.

File translated from TEX by TTH, version 3.40.
On 11 Aug 2004, 14:26.

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