Summer semester

Date

Speaker

Title

Abstract

2.03.2018

Adam Śpiewak

Absolute continuity of nonhomogeneous selfsimilar measures

During the talk I will present results from a recent preprint by
S. Saglietti, P. Shmerkin and B. Solomyak concerning properties of
nonhomogenous selfsimilar measures on the real line. The main result
states that the invariant measure for the system of the form $(\lambda_1 x +
t_1, \ldots, \lambda_k x + t_k)$ is absolutely continuous with respect to the
Lebesuge measure for almost every $(\lambda_1, \ldots, \lambda_k)$ with
similarity dimension strictly greater than one. The main difference with the
homogenous case ($\lambda_1 = \cdots = \lambda_k$) is lack of the convolution
structure of measures under consideration.

9.03.2018

Vasiliki Evdoridou

Commuting functions and the Fatou set

Let $f$ and $g$ be two rational functions that commute, i.e $f\circ g=g\circ f$.
Then it is known that $F(f)=F(g)$. We will discuss the analogous question for
the case where $f$ and $g$ are transcendental entire functions. We will see in
which cases we have that $F(f)=F(g)$ by presenting all the known results. We
will focus on a recent result by Benini, Rippon and Stallard and finally we
will discuss the remaining open question on this problem.

16.03.2018

Łukasz Pawelec

The differentiability of the hairs of $\exp(z)$

We will look at the paper by M. Viana da Silva, where the author proves that
the hair of the function $\lambda e^z$ is a smooth curve.

23.03.2018

Krzysztof Lech

Julia sets of random iterations of $z^2 + c_n$

The talk will consist of a presentation of a paper by Bruck, Buger
and Reitz. In said paper they describe necessary and sufficient conditions
for the connectedness of the Julia set of random iterations of the $z^2 + c_n$
family. We will also discuss what is known about conditions for the Julia
set being totally disconnected.

6.04.2018

Reza Mohammadpour Bejargafsheh

On the Hausdorff dimension of invariant measures of weakly contracting on
average measurable IFS

In this talk we give a contribution to the study of the multifractal
properties of measures which are invariant for iterated function systems. In a paper by
J. Myjak, T. Szarek the systems contracting on average and having
Dinicontinuous, separated from zero probabilities were considered and the
upper bound of the Hausdorff dimension of the unique invariant distribution
was given.
A. H. Fan, K. Simon, H. R. Toth, have shown Hausdorff dimension of any of
the possibly uncountably many invariant measures which the system is
contracting on average in a sense which is wide enough to permit the
existence of a common fix point at which some functions of the system are
expanding and perhaps none of them are contracting.
We are presenting an article by J. Jaroszewska and M. Rams who study iterated
function systems without uniqueness of invariant distributions in this
respect.

13.04.2018

Welington Cordeiro

$N$expansive homeomorphisms with the shadowing property

The dynamics of expansive homeomorphisms with the shadowing
property may be very complicated but it is quite well understood (see Aoki and
Hiraide’s monograph, for example). It is known that these systems admit only
a finite number of chain recurrent classes (Spectral Decomposition Theorem).
In 2012, Morales introduced a generalization of expansivity property, called
$N$expansive property. For every $N \in \mathbb N$, we will exhibit an
$N$expansive
homeomorphism, which is not $(N  1)$expansive, has the shadowing property
and admits an infinite number of chainrecurrent classes. We discuss some
properties of the local stable (unstable) sets of $N$expansive homeomorphisms
with the shadowing property and use them to prove that some types of the
limit shadowing property are present.

20.04.2018

Łukasz Chomienia

Selfsimilar sets, entropy and additive combinatorics

I will present the paper by Michael Hochman, in which author explains the method of the proof of the fact that measure of the degree of concentration of cylinders tends to zero superexponentially fast. At first, I will consider the discretization of the problem and I will solve it using combinatorial approach. Then I will back to the original question and I will try to imitate discrete case procedure. Of course, I will skip many technical details, especially in the continuous case.

27.04.2018

Rafał Tryniecki

Continuity of the Hausdorff measure of continued fractions and countable
alphabet iterated function system

I will be talking about a paper written by A. Zdunik and M. Urbanski about
Hausdorff measure of continued fractions. I will show that Hausdorff measure
of the set $J_n(G)$ of all numbers in [0,1], whose infinite continued fraction expansion
have all entries in finite set $\{1,\ldots,n\}$ satisfies $\lim_{n \to\infty}H_{h_n}(J_n(G)) = 1$, where $h_n$ is
the Hausdorff
dimension of $J_n(G)$, and $H_{h_n}$ is corresponding Hausdorff measure. I
will also show that this property is not too common, by constructing a class
of IFS, such that upper limit of Hausdorff measure in corresponding
Hausdorff dimension is smaller than the measure of the limit set.

4.05.2018

Bartłomiej Żak

Random interval transformations

During the third talk on random interval
homeomorphisms we analyze the case of zero Lapunov exponent at
point $0$. We show that there exists a neighbourhood of $0$ such that each
trajectory of the random walk defined by our random iterates of our
diffeomorphisms starting in this neighbourhood, leaves it in finite
time. Then, under the assumption that Lapunov exponent is greater than zero
at point $1$, we prove that in that case, for any neighbourhood of $0$,
expected time of leaving it is infinite and expected time of reentering it
is finite.

18.05.2018

Łukasz Treszczotko

Weak convergence to stable Lévy processes for nonuniformly hyperbolic dynamical systems

We consider invariance principles (functional limit theorems) in the domain of a stable law. The results are obtained by lifting the limit laws from an induced dynamical system to the original system. Examples include PomeauManneville intermittency maps.

25.05.2018

Jan Kwapisz

Eternal cosmological inflation, dynamical systems approach

Inflation started its career in the 80's when it turned out that classical
cosmology picture struggles with fine tuning of matter density i.e. very
uniform distribution of matter at the time of last scattering in circa 1084
disconnected regions. The Universe must have started its history with
very a special initial state. Guth and Starobinsky and argued that
the possible solution to this problem exists without finetuning the initial
conditions. If one assumes that at the very early stage of expansion of the
Universe there was an accelerated accelerated expansion era, which eventually went into the decelerated FLRW epoch, then the fine tuning problem is
solved. This solution of the problem is called inflation but it introduced an
issue of its own initial conditions what is under active investigation nowadays. There were many mechanisms proposed and discussed to generate
the $\ddot{a} > 0$ epoch just after the Big Bang.
In my talk I will discuss one special case: eternal
inflation within dynamical systems approach.

1.06.2018

Michał Rams

Invariant measures for a class of cocycles

We consider a step one cocycle of two homeomorphisms of interval
$[0,1]$ with the following properties: $f_0(0)=0$, $f_0(1)=1$, $f_0(x)>x$ for
$x\in (0,1)$ and $f_1(0)=1$, $f_1(1)=0$. We investigate the space of ergodic
invariant measures for this system. Clearly, it consists of two subsets:
measures supported on $\{0,1\}\times \Sigma_2$ and measures supported on
$(0,1)\times\Sigma_2$, and we want to know which measures in the former set
can be weak* aproximated by measures in the latter set. We will also
investigate the uniqueness of the measure of maximal entropy. This is a
joint work with L. Diaz, K. Gelfert, T. Marcarini.

8.06.2018

Ludwik Jaksztas

On the derivative of the Hausdorff dimension of the quadratic Julia
sets

Let $d(c)$ denote the Hausdorff dimension of the Julia set
$J(z^2+c)$. We will investigate the derivative $d'(c)$, for real $c$
converging to a parabolic parameter $c_0$. First, we will prove that $d'(c)$
tends to infinity, when $c\nearrow1/4$. Next, we will see that $d'(c)$ tends
to a constant or minus infinity depending on the value $d(c_0)$, where $c_0$
is a parabolic parameter with two petals.

15.06.2018

Piotr Gałązka

Hausdorff dimension of the escaping set for a family of meromorphic maps

Escaping set is a set of points which tend to infinity under iterates of a
map. During the talk we will see how large the Hausdorff dimension of the
escaping set is for maps from the family $R\circ \exp$, where $R$ is a
nonconstant rational map.
