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.

Reinforcement model on graphs and their limit behaviour

Speaker: 

Victor Kleptsyn

Institution: 

CNRS, Institut de Recherche Mathematique de Rennes

Time: 

Tuesday, October 30, 2018 - 1:00pm to 2:00pm

Host: 

Location: 

RH 440R

    The classical Polya urn process is a reinforcement process, in which there are balls of different color in the urn, we take out a ball at random, and the color that was just out of it gets an advantage for all future turns: we return this ball to the urn and add another one of the same color.   

    However, in this process on every step all the colors are competing. What will happen if on different steps there will be different subsets of competing colors? For instance, if there are companies that compete on different markets, or if a signal is choosing its way to travel?   

    Some questions here have nice and simple answers; my talk will be devoted to the results of our joint project with Mark Holmes and Christian Hirsch on the topic.

Three Fairy Math Stories

Speaker: 

D.Burago

Institution: 

Penn State University

Time: 

Thursday, October 25, 2018 - 2:00pm to 3:00pm

Location: 

RH 306

 Three different math stories in one lecture. Only definitions, motivations, results, some ideas behind proofs, open questions. 

1. One of the greatest achievements in Dynamics in the XX century is the KAM Theory. It says that a small perturbation of a non-degenerate completely integrable system still has an overwhelming measure of invariant tori with quasi-periodic dynamics. What happens outside KAM tori remains a great mystery. The main quantitate invariants so far are entropies.  It is easy, by modern standards, to show that topological entropy can be positive. It lives, however, on a zero measure set. We are now able to show that metric entropy can become positive too, under arbitrarily small C^{infty} perturbations, answering an old-standing problem of Kolmogorov. Furthermore, a slightly modified construction resolves another long–standing problem of the existence of entropy non-expansive systems. In these modified examples  positive metric entropy is generated in arbitrarily small tubular neighborhoods of one trajectory. Joint with S. Ivanov and Dong Chen.

2. A survival guide for a feeble fish and homogenization of the G-Equation. How fish can get from A to B in turbulent waters which maybe much fasted than the locomotive speed of the fish provided that there is no large-scale drift of the water in the ocean? This is related to the G-Equation and has applications to its homogenization. The G-equation is believed to govern many combustion processes, say wood fires or combustion in combustion engines (generally, in pre-mixed media with “turbulence".  Based on a joint work with S. Ivanov and A. Novikov.

3. Just 20 years ago the topic of my talk at the ICM was a solution of a problem which goes back to Boltzmann and has been formulated mathematically by Ya. Sinai. The conjecture of Boltzmann-Sinai states that the number of collisions in a system of $n$ identical balls colliding elastically in empty space is uniformly bounded for all initial positions and velocities of the balls. The answer is affirmative and the proven upper bound is exponential in $n$. The question is how many collisions can actually occur. On the line, one sees that  there can be $n(n-1)/2$ collisions, and this is the maximum. Since the line embeds in any Euclidean space, the same example works in all dimensions. The only non-trivial (and counter-intuitive) example I am aware of is an observation by Thurston and Sandri who gave an example of 4 collisions between 3 balls in $R^2$. Recently, Sergei Ivanov and me proved that there are examples with exponentially many collisions between  $n$ identical balls in $R^3$, even though the exponents in the lower and upper bounds do not perfectly match. Many open questions left.

Pages

Subscribe to RSS - Dynamical Systems