Winter semester

Date

Speaker

Title

Abstract

12.10.2018

Łukasz Pawelec

Quantitative recurrence

We will look for ways to improve upon the classic Poincare
recurrence lemma. Instead of just saying that almost every point returns to
its neighbourhood, we will ask e.g. how quickly it returns or what is the
distribution of the first return times. This topic has yielded numerous
papers over the last years. I want to start with the most basic results and
continue into the overview of the field ending with some of the newest
results.

19.10.2018

Krzysztof Lech

Geometric properties of Julia sets of the composition of
polynomials $z^2 + c_n$

We will discuss a paper by Bruck about geometric properties of
Julia sets of random quadratic polynomials. In the paper he proves that the
Julia set is a quasicircle for a certain range of parameters $c_n$,
corresponding to the range for which it is a quasicircle in the autonomous
case. Moreover he gives several results concerning the Lebesgue measure and
Hausdorff dimension of these Julia sets, which we shall also present.

26.10.2018

Jakub Andruszkiewicz

Hausdorff dimension of the set of badly approximable irrationals

We will show that the numbers which are badly aproximated by
the convergents of their continued fractions are the ones with bounded set of the continued fraction coefficients.
Then we will prove some estimates for the Hausdorff dimension
of the set of numbers with coefficients bounded by some fixed number.
We will use that to show that the Hausdorff dimension of the set of numbers
with bounded coefficients is $1$ (although its Lebesgue measure is $0$).
The talk follows the proofs in A. Kryska's BSc thesis, based on V. Jarnik's articles.

9.11.2018

Adam Król

Line, spiral, dense

Following Neil Dobbs, we show that the image of a typical straight line under two iterates of exponential map is dense in the plane.

16.11.2018

Patrick Comdühr (ChristianAlbrechtsUniversität zu Kiel)

Nowhere differentiable hairs for entire maps

In 1984 Devaney and Krych showed that for the exponential family $\lambda e^z$, where $0<\lambda <1/e$, the Julia set consists of uncountably many pairwise disjoint simple curves tending to infinity, which they called hairs. Viana proved that these hairs are smooth. In contrast to Viana's result we construct an entire function, where the Julia set consists of hairs, which are nowhere differentiable.

23.11.2018

Núria Fagella (Universitat de Barcelona)

Periodic cycles and singular values of entire transcendental
functions

Location of periodic points of a dynamical system, and local dynamics around
them are classical problems. Among the nonrepelling periodic points we find
the equilibria of the system. When the iterated function is entire
(polynomial or transcendental) the Separation Theorem describes the
distribution of the nonrepelling periodic points with respect to the
"external rays", an invariant set of points in the escaping set. In this
talk we shall relate these objects also to the singular values of the
function, i.e. points where the map fails to be a local homeomorphisms.
These results give an alternative proof of the well known FatouShishikura
inequality which bounds the number of nonrepelling cycles in terms of the
number of singular values of the map.

30.11.2018

Rafał Tryniecki

Exact Hausdorff measure and intervals of maximum
density for Cantor sets

We will analyze a paper by E. Ayer and S. Strichartz about
Hausdorff measure and maximum density of Cantor sets. First of all, we will
consider results about density in case where "islands" do not touch, and
show where the maximum is obtained. Then we will allow touching "islands"
and show that under certain condition we still can find where is the maximum
density obtained. Next, we will see that without this condition, usually the
supremum of the density is not reached. Finally, we will show that results
shown in this paper do not generalise to linear selfsimilar sets in higher
dimentional Euclidean spaces.

7.12.2018

Reza Mohammadpour Bejargafsheh

Dimension product structure of hyperbolic set

We will show Hausdorff dimension for Smale solenoids. Moreover we will
show that Hausdorff dimension of all stable slices are the same. The talk
follows a paper by Hasselblat and Schmeling.

14.12.2018

Leticia PardoSimón (University of Liverpool)

Escaping dynamics of a class of transcendental functions

As a partial answer to Eremenko's conjecture, it is known for
functions with bounded singular set and of finite order that every point in
their escaping set can be connected to infinity by an escaping curve. Even
if those curves, called "hairs" or "rays" not always land, this has been
positively proved for some functions with bounded postsingular set by
showing that their Julia set is structured as a Cantor Bouquet. In this talk
I will consider certain functions with bounded singular set but unbounded
postsingular set whose singular orbits escape at some minimum speed. In this
setting, some hairs will split when they hit critical points. We show that
the existence of a map on their parameter space whose Julia set is a Cantor
Bouquet guarantees that such hairs, if maybe now with split ends, still
land.

21.12.2018

Jacek Rutkowski

Brennan's conjecture and the Mandelbrot set

We prove that Brennan's conjecture is satisfied for some particular class of domains.
Special character of this class causes that the proof will amount to some geometrical observation about the Mandelbrot set.
The talk will follow the paper by Barański, Volberg and Zdunik.

11.01.2019

Adam Śpiewak

Maximising Bernoulli measures and dimension gaps for countable branched systems

It is known since a paper by Kifer, Peres and Weiss that
$\dim \mu$ is uniformly bounded away from $1$ among all probability measures
$\mu$ on the unit interval which make the digits of the continued fraction
expansion i.i.d. random variables. I will present a recent paper
"Maximising Bernoulli measures and dimension gaps for countable branched
systems" by Simon Baker and Natalia Jurga, where the authors prove that
there exists a measure maximising the dimension in this class. The
results are valid for a more general class of countable branched systems.

18.01.2019

Welington Cordeiro

Beyond topological hyperbolicity

We generalize the usual notion of topological hyperbolicity introducing the concept continuumwise hyperbolicity. We discuss examples of these
systems and characterize the possible dynamical phenomena that can occur on
cwhyperbolic transitive homeomorphisms: either they are topologically hyperbolic, or there exist arbitrarily small dynamical balls containing topological
semihorseshoes, that are periodic sets where the dynamics is semiconjugated to the shift of two symbols. We prove cwhyperbolicity implies some of the
standard properties of hyperbolic systems, such as the shadowing property and
finiteness of chain recurrent classes. Work in progress with Alfonso Artigue,
Bernardo Carvalho and José Vieitez.

25.01.2019

Łukasz Treszczotko

Weak convergence to extremal processes for some dynamical systems

For a measure preserving dynamical system $(X, f, \mu)$ we consider the
time series of maxima $M$ associated to the process generated by some observable
$\varphi : X \to \mathbb R$. Using a point process approach we establish weak convergence of
the rescaled process $M$ to an extremal process.
