# Ergodic Schrodinger operators in the infinite-measure setting

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

Victor Kleptsyn

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

Victor Kleptsyn

## Institution:

CNRS, University of Rennes 1

## Time:

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

## 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

Yulij Ilyashenko

## Institution:

Independent University of Moscow, Higher School of Economics

## Time:

Tuesday, February 12, 2019 - 1:00pm

## 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.