# Abstracts for Dynamics and Numbers

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## A generalisation of the Wiener-Wintner theorem

Posted in
Speaker:
Tanja Eisner
Zugehörigkeit:
U Leipzig\MPI
Datum:
Die, 2014-06-03 14:00 - 15:00
Location:
MPIM Lecture Hall

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

Posted in
Speaker:
Alexander Blokh
Organiser(s):
U Alabama\MPI
Datum:
Mit, 2014-06-04 14:15 - 15:15
Location:
MPIM Lecture Hall
Parent event:
Dynamics and Numbers
Parent event:
Number theory lunch seminar

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

Posted in
Speaker:
Michal Misiurewicz
Zugehörigkeit:
IUPUI\MPI
Datum:
Fre, 2014-06-06 14:00 - 15:00
Location:
MPIM Seminar Room
Parent event:
Dynamics and Numbers

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

Posted in
Speaker:
Piotr Oprocha
Zugehörigkeit:
AGH U\MPI
Datum:
Mon, 2014-06-16 14:00 - 15:00
Location:
MPIM Lecture Hall
Parent event:
Dynamics and Numbers

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

Posted in
Speaker:
Jian Li
Zugehörigkeit:
Shantou U\MPI
Datum:
Mon, 2014-06-16 15:00 - 16:00
Location:
MPIM Lecture Hall
Parent event:
Dynamics and Numbers

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

Posted in
Speaker:
Tomasz Downarowicz
Organiser(s):
U Tech Wroclaw\MPI
Datum:
Die, 2014-06-17 15:00 - 16:00
Location:
MPIM Lecture Hall
Parent event:
Dynamics and Numbers

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

Posted in
Speaker:
Lubomir Snoha
Zugehörigkeit:
Matej Bel U\MPI
Datum:
Mit, 2014-06-18 15:00 - 16:00
Location:
MPIM Lecture Hall
Parent event:
Dynamics and Numbers

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

Posted in
Speaker:
Bartłomiej Bzdęga
Zugehörigkeit:
Poznan
Datum:
Mit, 2014-06-18 16:30 - 17:30
Location:
MPIM Lecture Hall
Parent event:
Dynamics and Numbers
Parent event:
Number theory lunch seminar

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

Posted in
Speaker:
Henk Bruin
Zugehörigkeit:
U Wien\MPI
Datum:
Fre, 2014-06-20 14:00 - 15:00
Location:
MPIM Lecture Hall
Parent event:
Dynamics and Numbers

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

Posted in
Speaker:
Dominik Kwietniak
Zugehörigkeit:
Jagiellonian U\MPI
Datum:
Fre, 2014-06-20 15:30 - 16:30
Location:
MPIM Lecture Hall
Parent event:
Dynamics and Numbers

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. .

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