Alternatively have a look at the program.

## A generalisation of the Wiener-Wintner theorem

We discuss a nilsequence version of the classical Wiener-Wintner theorem on convergence of weighted ergodic averages due to Host and Kra and present a uniform version of this result. This is a joint work with Pavel Zorin-Kranich.

## Combinatorial models for spaces of cubic polynomials

To construct a model for a connectedness locus of polynomials of degree $d\ge 3$ (cf with Thurston's model of the Mandelbrot set), we define *linked* geolaminations $\mathcal{L}_1$ and $\mathcal{L}_2$. An *accordion* is defined as the union of a leaf $\ell$ of $\mathcal{L}_1$ and leaves of $\mathcal{L}_2$ crossing $\ell$. We show that any accordion behaves like a gap of one lamination and prove that the maximal *perfect* (without isolated leaves) sublaminations of $\mathcal{L}_1$ and $\mathcal{L}_2$ coincide.

## No semiconjugacy to a map of constant slope

We study countably piecewise continuous, piecewise monotone interval

maps. We establish a necessary and sufficient criterion for the

existence of a nondecreasing semiconjugacy to a map of constant slope

in terms of the existence of an eigenvector of an operator acting on a

space of measures. Then we give sufficient conditions under which this

criterion is not satisfied. Finally, we give examples of maps not

semiconjugate to a map of constant slope via a nondecreasing map. Our

## Return times and synchronous recurrence

Let $(X,f)$ be a discrete dynamical system and let $\mathcal{F}$ be a hereditary upward set of subsets of $\mathbb{N}$. A point $x$ is $\mathcal{F}$-recurrent, if for any open neighborhood $U$ of $x$, return times of $x$ to $U$ are in $\mathcal{F}$, that is $\{n : f^n(X)\}\in \mathcal{F}$. A point $x$ is $\mathcal{F}$-PR if for any $\mathcal{F}$-recurrent point $y$ in any dynamical system $(X,g)$ the pair $(x,y)$ is recurrent for $(X\times Y, f\times g)$. In this talk we will present recent results and open problems related to the $\mathcal{F}$-PR property.

## Mean equicontinuity and mean sensitivity

In this talk, we study equicontinuity and sensitivity in the mean sense.

We show that every ergodic invariant measure of a mean equicontinuous

(i.e. mean-L-stable) system has discrete spectrum. Dichotomy results related

to mean equicontinuity and mean sensitivity are obtained when a dynamical

system is transitive or minimal. Localizing the notion of mean equicontinuity,

notions of almost mean equicontinuity and almost Banach mean equicontinuity

are introduced. It turns out that a system with the former property may have

## New approach to entropy of countable group actions

There exists an astonishingly simple formula for the dynamical entropy of a process

(measure-preserving transformation with a partition),which can be easily extended

to actions of any countable group. This formula coincides with the standard one up

to amenable groups. I will discuss its applicability to other groups and, exploring

another direction, to topological entropy.

## Uniquely minimal spaces

We construct a continuum X such that it admits a minimal homeomorphism T

and the only self-homeomorphisms of X are the iterates of T. This is a joint

work with T. Downarowicz and D. Tywoniuk.

## Ternary Cyclotomic Polynomials

Cyclotomic polynomials of the form Phi_{pqr}(x) with p,q,r distinct odd primes are called ternary

cyclotomic polynomials. They are the easiest class of polynomials for which the behaviour of the

coefficients is not very well understood. I will present my methods and ideas to evaluate coefficients

and differences between consecutive coefficients of such polynomials.

## Substitution shifts and renormalization for potentials

Motivated by Bowen's theory of thermodynamic formalism of

subshifts of finite type, Hofbauer started to create symbolic models with

non-Hoelder potentials as simple examples exhibiting phase transitions.

Hofbauer's examples relate directly to the Pomeau-Manneville map, which

Baraviera-Leplaideur-Lopes related again to a particular

substitution-based renormalization operator. In this talk, I want to

report on joint work with Leplaideur how this scheme extends to

non-trivial substitutions (Thue-Morse and Fibonacci), and discuss the

## Entropy and independence in symbolic dynamics with connections to number theory

We introduce a new family of shift spaces - the subordinate

shifts. Using them we prove in an elementary way that for every

nonnegative real number t one can find a shift space with entropy t.

Moreover, we show that there is a connection between positive entropy

and combinatorial independence of a shift space. Positive entropy can

be characterized through existence of a large (in terms of asymptotic

and Shnirelman densities) set of coordinates along which the highest

possible degree of randomness in points from the shift is observed. .

© MPI f. Mathematik, Bonn | Impressum & Datenschutz |