3.1.2 Edge and Bulk Currents in the Integer Quantum Hall Effect

Jeffrey Schenker

Institute for Advanced Study jeffrey@ias.edu

Two apparently different conductances σBulk and σEdge have been used to explain the integer quantum Hall effect, depending on whether the currents in the sample are ascribed to the bulk or the edge. The bulk conductance σBulk , as expressed through a linear response formula, is well defined when the Fermi energy falls in a mobility gap, that is a band of localized states. However, the edge conductance σEdge, expressed as the derivative of the steady state edge current with respect to the Fermi energy, is ill defined unless the Fermi energy falls in true gap. A physically suitable expression for σEdge can be obtained from a modified formula involving either

(1) a truncated trace and a correction term or (2) time averaging. With this modified expression the equality σEdge = σBulk is a theorem, as expected from heuristic arguments. (Joint work with A. Elgart and G.M. Graf)

3.1.3 Quantum Phases of Cold Bosons in Optical Lattices

Jakob Yngvason

Universität Wien yngvason@thor.thp.univie.ac.at

In recent years it has become possible to trap ultracold atoms and molecules in lattices generated by laser beams (optical lattices). By varying the experimentally tunable parameters transitions between various phases of the trapped gas, in particular between a Bose Einstein condensate and a Mott insulator phase, can be produced. The talk reviews these developments, and rigorous theoretical results on such transitions, obtained in collaboration with M. Aizenman, E.H. Lieb, R. Seiringer and J.P. Solovej, will be presented.

3.2 Dynamical Systems

Organizers W. de Mello (Rio de Janeiro), F. Ledrappier (Notre Dame)

3.2.1 Statistical Stability for Hénon Maps of Benedics-Carleson Type

Jose Ferreira Alves

University of Porto jfalves@fc.up.pt

We consider the two-parameter family of Hénon maps in the plane (x, y) → (1 − ax2 + y, bx). Benedicks and Carleson proved that there is a positive Lebesgue measure set A of parameters (a, b) for which the corresponding Hénon map has a chaotic attractor. Subsequent work by Benedicks and Young showed that each of these attractors supports an SRB measure, i.e. a probability measure which describes the statistics of Lebesgue almost every point in a neighborhood of the attractor. Here we show that the SRB measures vary continuously in weak* topology with the parameters (a, b) A. This is a joint work with M. Carvalho and J.M. Freitas.

3.2.2 Entropy and the Localization of Eigenfunctions

Nalini Anantharaman

ENS-Lyon

Nalini.Anantharaman@umpa.ens-lyon.fr

We study the large eigenvalue limit for eigenfunctions of the Laplacian, on a compact negatively curved manifold. According to the Quantum Unique Ergodicity conjecture, eigenfunctions must become equidistributed in phase space, meaning that the Wigner transforms of eigenfunctions must converge weakly to the Liouville measure. We find a positive lower bound for the Kolmogorov-Sinai entropy of limits of these Wigner measures, which shows that eigenfunctions must be delocalized to a certain extent. Part of this work is joint with Stephane Nonnenmacher (CEA Saclay).

3.2.3The Spectrum of the Almost Mathieu Operator in the Subcritical Regime

Artur Ávila

CNRS-Jussieu artur@math.sunysb.edu

We discuss the almost Mathieu operator H : l2(Z) → l2(Z), (H u)n

un−1 + 2λ cos 2π(θ + nα), where λ > 0 (the coupling), α R \ Q (the frequency), and θ R (the phase) are parameters. The nature of the spectral measures has been subject of several conjectures since 1980, when Aubry-André proposed the following picture:

1-Localization (point spectrum with exponentially decaying eigenfunctions) for the supercritical regime λ > 1,

2-Absolutely continuous spectrum for the subcritical regime λ < 1, both regimes being linked by Aubry duality.

Localization turns out to be very sensitive to arithmetics (and fails generically), so the description of the supercritical regime could only be proved in the “almost

every” sense. Whether something similar happened in the subcritical regime remained unclear. We will discuss recent progress towards the complete solution of this problem.

3.2.4 Hyperbolicity Through Entropy

Jerome Buzzi

École Polytechnique buzzi@math.polytechnique.fr

We show how (robust) entropy assumptions yields (what we call) semi-uniform hyperbolic structures which allow the global analysis of some classes of smooth dynamical systems from the point of view of their complexity. These classes include coupled interval maps with positive entropy and models for surface diffeomorphisms.

3.2.5 Robust Cycles and Non-dominated Dynamics

Lorenzo J. Diaz

PUC – Rio de Janeiro lodiaz@mat.puc-rio.br

The Newhouse’s construction of C2-surface diffeomorphisms having a hyperbolic sets with robust tangencies relies on the notion of thick hyperbolic set. These thick hyperbolic sets are the key for so-called coexistence phenomenon (existence of locally residual sets of diffeomorphisms having simultaneously infinitely many sinks and sources). These constructions are typically C2. The goal of this talk is to discuss similar phenomena in higher dimensions and in the C1-topology.

We first explain the generation of robust cycles in the C1-topology and obtain some dynamical consequences from this fact. We also discuss the role of the robust cycles for generating robustly non-dominated dynamics and deduce some strong forms of the coexistence phenomenon from the lack of domination.

3.2.6 Hyperbolicity in One Dimensional Dynamics

Oleg Kozlovskiy

Warwick Mathematics Institute oleg@maths.warwick.ac.uk

Recently together with W. Shen and S. van Strien we were able to prove dencity of hyperbolicity for all real one dimensional maps and also for a large class of one dimensional holomorphic maps. During the talk we will discuss these results together with other recent developments in the subject.