Hausdorff Dimension of the spectrum of the Thue-Morse Hamiltonian

Speaker: 

William Wood

Institution: 

UC Irvine

Time: 

Tuesday, November 19, 2019 - 1:00pm to 2:00pm

Location: 

RH 440R

We will discuss a dynamical approach by Qinghui Liu and Yanhui Qu to study the spectrum of the Thue-Morse Hamiltonian, a Schrodinger operator on l^2. They use the trace map formalism to describe the spectrum, and study the corresponding polynomials to characterize subsets of their zeros that converge to a Cantor set. In doing this they establish bounds of the Hausdorff dimension. We will also discuss a possiblility to obtain similar bounds on the thickness of the spectrum. 

Renormalization: from holomorphic dynamics to two dimensions

Speaker: 

Denis Gaidashev

Institution: 

Uppsala University

Time: 

Tuesday, November 12, 2019 - 1:00pm to 2:00pm

Host: 

Location: 

RH 440R

I will describe recent advances in promoting renormalization techniques from one dimensional holomirphic setting to two dimensions, together with the set of questions that these techniques are aiming to address, such as rigidity and geometry of attractors.

Cantor Sets, their Sums, and the Lagrange and Markov Spectra

Speaker: 

William Wood

Institution: 

UC Irvine

Time: 

Tuesday, October 29, 2019 - 1:00pm to 2:00pm

Location: 

RH 440R

Cantor sets and their fractional dimensions are related to many unsolved conjectures. One of these conjectures, known as Palis' Conjecture, involves the dimension of the Minkowski sums of Cantor sets, and in this talk we will describe the details of the Palis Conjecture. In addition, we will use some of the tools involving Cantor sets to describe of the some properties of the Lagrange and Markov spectra, in particular, related to existence of the Hall's ray. The talk will be mostly based on the work done by Moreira in recent years.

Ergodic Schrodinger operators in the infinite-measure setting

Speaker: 

Jake Fillman

Institution: 

Texas State University

Time: 

Thursday, October 24, 2019 - 2:00pm to 3:00pm

Location: 

RH 340P

We develop the basic spectral theory of ergodic Schrodinger operators when the underlying dynamics are given by a conservative ergodic transformation of a \sigma-finite measure space. Some fundamental results, such as the Ishii--Pastur theorem carry over to the infinite-measure setting. We also discuss some examples in which straightforward analogs of results from the probability-measure case do not hold. We will discuss some examples and some interesting open problems. 

The talk is based on a joint work with M. Boshernitzan, D. Damanik, and M. Lukic.

Nonstationary low-dimensional dynamics

Speaker: 

Victor Kleptsyn

Institution: 

IRM de Rennes

Time: 

Thursday, September 26, 2019 - 2:00pm to 3:00pm

Location: 

RH 340P

We consider discrete Schr\"odinger operators on $\ell^2(\mathbb{Z})$ with bounded random but not necessarily identically distributed values of the potential. The distribution at a given site is not assumed to be absolutely continuous (or to contain an absolutely continuous component). We prove spectral localization (with exponentially decaying eigenfunctions) as well as dynamical localization for this model.

An important ingredient of the proof is a non-stationary analog of the Furstenberg Theorem on random matrix products,
which is also of independent interest. 

This is a joint project with A.Gorodetski.

Skew products with the one-dimensional fiber

Speaker: 

Victor Kleptsyn

Institution: 

CNRS, University of Rennes 1

Time: 

Tuesday, February 26, 2019 - 1:00pm to 2:00pm

Host: 

Location: 

RH 440R

My (survey) talk will be devoted to different examples (by many authors) of dynamics with one-dimensional 
fiber. Such examples (intermingled attractors, bony attractors, Fubini nightmare, zero Lyapunov exponents, 
etc.) are often quite geometric and can be easily visualized, and at the same time manifest a highly 
nontrivial behavior.

New fractal set "Bubbles" and complex rotation numbers

Speaker: 

Yulij Ilyashenko

Institution: 

Independent University of Moscow, Higher School of Economics

Time: 

Tuesday, February 12, 2019 - 1:00pm

Host: 

Location: 

RH 440R

About 40 years ago Arnold suggested  a  construction of so called  complex rotation numbers. Consider a circle diffeomorphism, lifted to a map of the real line that commutes with the shift by 1. For each complex number in the  open upper half-plane consider the lifted map combined with the shift by this number. Glue the  real line and the shifted real line via this map  and factorize the cylinder obtained by the unit shift. Consider the modulus of the elliptic curve thus obtained. It is sometimes called the complex rotation number of the original map combined with the shift. We get the moduli map of the upper half plane into itself.  The questions due to Arnold and Ghys can be formulated as follows:

How the boundary values of the moduli map are related to the rotation number of the original shifted map considered as a function on the absolute?

It appeared that for generic circle diffeomorphisms the boundary values of the moduli map form a fractal set ``Bubbles'' located inside the upper half plane: any rational point on the real axis belongs to a closed curve from this fractal set; this curve is  called a bubble. The boundary values of the moduli map coincide with  the rotation number when these values are irrational. The way of approximate calculation of bubbles is suggested, their possible bizarre shapes are drawn, and it is proved that the bubbles can intersect and self-intersect.

Self-similarity of the fractal set ``bubbles'' is established.

Recently it was  found out that the fractal set ``bubbles'' does not depend continuously on the original circle diffeomorphism. This drastically distinguishes it from Arnold tongues that depend on the original diffeomorphism continuously.

Most part of these results belongs to N. Goncharuk; other part is obtained by Buff, Moldavskis, Rissler and others.

Erratic behavior for one-dimensional random walks in a generic quasi-periodic environment

Speaker: 

Maria Saprykina

Institution: 

KTH, Sweden

Time: 

Monday, December 3, 2018 - 11:00am to 12:00pm

Host: 

Location: 

RH 420

Consider a Markov chain on a one-dimensional torus $\mathbb T$, where a moving point jumps from $x$ to $x\pm \alpha$ with probabilities $p(x)$ and $1-p(x)$, respectively, for some fixed function $p\in C^{\infty}(\mathbb T, (0,1))$ and $\alpha\in\mathbb R\setminus \mathbb Q$. Such Markov chains are called random walks in a quasi-periodic environment. It was shown by Ya. Sinai that for Diophantine $\alpha$ the corresponding random walk has an absolutely continuous invariant measure, and the distribution of any point after $n$ steps converges to this measure. Moreover, the Central Limit Theorem with linear drift and variance holds.

In contrast to these results, we show that random walks with a Liouvillian frequency $\alpha$ generically exhibit an erratic statistical behavior. In particular, for a generic $p$, the corresponding random walk does not have an absolutely continuous invariant measure, both drift and variance exhibit wild oscillations (being logarithmic at some times and almost linear at other times), Central Limit Theorem does not hold.

These results are obtained in a joint work with Dmitry Dolgopyat and Bassam Fayad.

Parametric Furstenberg Theorem and 1D Anderson Localization

Speaker: 

Victor Kleptsyn

Institution: 

CNRS

Time: 

Friday, November 2, 2018 - 2:00pm to 3:00pm

Host: 

Location: 

RH 340P

We consider random products of SL(2, R) matrices that depend on a parameter in a non-uniformly hyperbolic regime. We show that if the dependence on the parameter is monotone then almost surely the random product has upper (limsup) Lyapunov exponent that is equal to the value prescribed by the Furstenberg Theorem (and hence positive) for all parameters, but the lower (liminf) Lyapunov exponent is equal to zero for a dense $G_\delta$ set of parameters of zero Hausdorff dimension. As a byproduct of our methods, we provide a purely geometrical proof of Spectral Anderson Localization for discrete Schrodinger operators with random potentials (including the Anderson-Bernoulli model) on a one dimensional lattice. This is a joint project with A.Gorodetski.

Pages

Subscribe to RSS - Dynamical Systems