Seminars & Events for Ergodic Theory & Statistical Mechanics
Effect of Emergent Distinguishability of Particles in a nonEquilibrium Chaotic System
We consider the behavior of classical particles which evolution consists of free motion interrupted by binary collisions. The fluid of hard balls and the dilute gas with arbitrary shortrange interactions are treated, where the total number of particles is moderate (say, five particles). We assume that the decay of correlations, characteristic for chaotic systems, holds (it can be considered proved for hard balls). We show that the numbers of collisions of a given particle with other particles grow effectively as a biased random walk. This is used to prove that over indefinitely long periods of time each particle has preferences: it systematically collides more with certain particles and less with others. Thus certain particles are effectively attracted and certain others are repelled, making the particles effectively distinguishable.
On a Hidden Symmetry of Simple Harmonic Oscillators
Since the original 1926 Schroedinger's paper, there was a misconception that the “simple” harmonic oscillator can be solved only by the separation of variables, which results in a traditional “static” electron density distribution.
It is not entirely accurate and a nontrivial oscillator hidden symmetry group, found by Niederer in 1973, provides "dynamic solutions". The phase space oscillations of the electron position and linear momentum probability distributions are computer animated and some possible applications to quantum optics are briefly discussed.
A visualization of the Heisenberg Uncertainty Principle is presented.
Effective bisector estimate for PSL(2,C) with applications to circle packings
Let Gamma be a nonelementary discrete geometrically finite subgroup of PSL(2,C). Under the assumption that the critical exponent of Gamma is greater than 1 we prove an effective bisector counting theorem for Gamma. We then apply this Theorem to the Apollonian circle packing problem to get power savings and to compute the overall constant. The proof relies on spectral theory of Gamma\PSL(2,C).
Noncollision Singularities in Planar Twocentertwobody Problem
In this work we study a model called planar 2center2body problem. In the plane, we have two fixed centers Q_1=(\chi,0), Q_2=(0,0) of masses 1, and two moving bodies Q_3 and Q_4 of masses \mu. They interact via Newtonian potential. Q_3 is captured by Q_2, and Q_4 travels back and forth between two centers. Based on a construction of Gerver, we prove that there is a Cantor set of initial conditions which lead to solutions of the Hamiltonian system whose velocities are accelerated to infinity within finite time avoiding all early collisions. We consider this model as a simplified model for the planar fourbody problem case of the Painleve conjecture.
This is a joint work with Dmitry Dolgopyat.
Minimal Interval Exchange Transformations with Many Ergodic Measures
We will give a brief survey of interval exchange transformations, especially results on minimality and unique ergodicity. In particular, we shall prove that a known upper bound on the number distinct ergodic probability measures for a minimal IETs is sharp. This is a generalization of known examples for IETs with orderreversing permutations.
The Spectrum of Random Kernel Matrices
We consider $nbyn$ matrices whose $(i, j)th$ entry is $f(X_i^T X_j)$, where $X_1, ...,X_n$ are i.i.d. standard Gaussian random vectors in $R^p$, and the kenrel function f is a realvalued function. We study the weak limit of the spectral density when p and n go to infinity and $p/n = \gamma$ which is a constant. The limiting spectral density is dictated by a cubic equation involving its Stieltjes transform, and the parameters of the cubic equation are decided by the Hermite expansion of the rescaled kernel function. While the case of kernel functions that are differentiable at the origin has been previously resolved by ElKaroui (2010), our result is applicable to nonsmooth kernel functions, e.g. the sign function.
Local semicircle law in the bulk for Gaussian $β$ensemble
Paper available at: http://arxiv.org/abs/1112.2016v1
Badly approximable directions on flat surfaces and bounded geodesics in Teichmueller space
In this talk we show that directions on flat surfaces which are poorly approximated by saddle connection directions are a winning set for Schmidt's game. This extends a result of Schmidt for the torus and strengthens a result of Kleinbock and Weiss. It is equivalent to saying that the Teichmueller geodesic flow is bounded. We go on to show that the set of bounded geodesics is winning as a subset of projective measured laminations, answering a question of McMullen. This is joint work with Yitwah Cheung and Howard Masur.
Invariant Measures, Conjugations and Renormalizations of Circle Maps with Break points
An important question in circle dynamics is regarding the absolute continuity of an invariant measure. We will consider orientation preserving circle homeomorphisms with break points, that is, maps that are smooth everywhere except for several singular points at which the first derivative has a jump. It is well known that the invariant measures of sufficiently smooth circle diffeomorphisms are absolutely continuous w.r.t. Lebesgue measure. But in the case of homeomorphisms with break points the results are quite different. We will discuss conjugacies between two circle homeomorphisms with break points. Consider the class of circle homeomorphisms with one break point $b$ and satisfying the KatznelsonOrnsteins smoothness condition i.e. $Df$ is absolutely continuous on $[b, b + 1]$ and $D^2f \in L^p(S^1, dl)$, $p > 1$.
Reducibility for the quasiperiodic linear Schrödinger and wave equations
We shall discuss reducibility of these equations on the torus with a small potential that depends quasiperiodically on time. Reducibility amounts to "reduce" the equation to a timeindependent linear equation with pure point spectrum in which case all solutions will be of Floquet type. For the Schrödinger equation, this has been proven in a joint work with S. Kuksin, and for the wave equation we shall report on a work in progress with B. Grebert and S. Kuksin.  


Effective discreteness of the 3dimensional Markov spectrum
Let the set O={nondegenerate, indefinite, real quadratic forms in 3variables with determinant 1}. We define for every form Q in the set O, the Markov minimum m(Q)=min{Q(v): v is a nonzero integral vector in $R^3$}. The set M={m(Q): Q is in O} is called the 3dimensional Markov spectrum.
An early result of CasselsSwinnertonDyer combined with Margulis' proof of the Oppenheim conjecture asserts that, for every a>0 {M \intersect (a, \infty)} is a finite set. In this lecture we will show that
#{M \intersect (a, \infty)}<< a^{26}.
This is a joint work with Prof. Margulis, and our method is based on dynamics on homogeneous spaces.
Perturbations of geodesic flows producing unbounded growth of energy
We consider a geodesic flow on a manifold endowed with some generic Riemannian metric. We couple the geodesic flow with a timedependent potential driven by an external dynamical system, which is assumed to satisfy some recurrence condition. We prove that there exist orbits whose energy grows unboundedly at a linear rate with respect to time; this growth rate is optimal. In particular, we obtain unbounded growth of energy in the case when the external dynamical system is quasiperiodic, of rationally independent frequency vector (not necessarily Diophantine). Our result generalizes Mather's acceleration theorem and is related to Arnold's diffusion problem. It also extends some earlier results by Delshams, de la Llave and Seara.
Moment estimates for squarefree integers on short intervals
Squarefree integers are known to have asymptotic density 6/(pi^2). Fix some x and let n be distributed uniformly on the integers between 1 and x. Consider the corresponding variance of the number of squarefree integers on a short interval [n+1, n+N] and let x tend to infinity. In 1982, R.Hall proved that the limiting variance behaves asymptotically, as N tends to infinity, like C*N^{1/2} for some constant C. In 1987, Hall also derived some estimates for higher moments of this random variable. Following another method, we will obtain a different estimate for the third moment. If time permits, we will also discuss higher moments and generalization of Hall's result to the case of kfree integers.