Summer semester

Date

Speaker

Title

Abstract

28.02.2020

Anna Kryska

Schmidt's game on certain fractals

We will describe Schmidt's game played on the support of absolutely friendly
measure and present its basic properties. In particular, we will show how to
compute the Hausdorff dimension for some winning sets.

6.03.2020

Michał Lemańczyk

Multiplicative convolution, entropy, onesdensity and lack of the
Gibbs property

I will consider subshifts $(X, S)$, where $X \subset \{0, 1\}^\mathbb Z$ and $S$ denotes the left shift. Given an ergodic measure $\nu$ on $(X, S)$ I will be interested in the measure $\kappa$ of the form $\kappa = \nu * B$ (multiplicative convolution measure), where $B$ stands for the Bernoulli measure on $\{0, 1\}^\mathbb Z$ with $B([0]) = B([1]) = 1/2$ and $\nu * B$ is the image of the product measure $\nu \otimes B$ under coordinatewise multiplication. I will provide some criterion under which $\kappa$ does not satisfy the Gibbs property. The criterion will be
expressed in terms of the density of ones. This talk will be based on the joint work with Joanna KułagaPrzymus.

13.03.2020

Mareike Wolff (ChristianAlbrechtsUniversität zu Kiel)

Exponential polynomials with Fatou and nonescaping sets of finite area

Let us consider entire functions of the form
$f(z)=\sum_{j=1}^NQ_j(z)\exp(b_jz^d+P_j(z))$, where $d\ge3$, and
$P_j,Q_j$ are polynomials with $\deg(P_j) < d$. Under suitable assumptions
on the numbers $b_j$, the Fatou set and the complement of the socalled
fast escaping set of $f$ have finite area. We will discuss these results
and sketch the proof.

27.03.2020

Krzysztof Lech

Random quadratic Julia sets and quasicircles

We will discuss some results by Rainer Brück concerning
nonautonomous Julia sets of quadratic maps $z^2 + c_n$. For $c_n$ chosen from a
disk of radius 1/4 the Julia set is a quasicircle, which is a generalization
of the analogous autonomous result. The proof of the above will be
presented, among some other related results.

3.04.2020

Rafał Tryniecki

Continuity of Hausdorff measure

We will discuss results of continuity of Hausdorff measure of Continued
Fractions by M. Urbanski and A. Zdunik. The main part of the proof will be
presented, as well as problems and results in analogous case where the IFS
consists of similarities.

8.04.2020 (Wednesday)

Piotr Rutkowski

Random iteration of Moebius transformations and Furstenberg's theorem

We will specify conditions under which for almost every sequence of
iid random maps on $SL(2,\mathbb R)$ the orbit of a nonzero initial point in $\mathbb R^2$
tends to infinity exponentially fast. Later, we will translate this statement
into the setup of Moebius transformations on the upper halfplane.

17.04.2020

Zofia Grochulska

Volume preserving homemorphisms of the cube

In this talk I will present a method of discrete approximation of
volume (i.e. measure) preserving homeomorphisms together with its
applications. We will show Peter Lax's result that one can find a
permutation of small dyadic cubes arbitrarily close to such a mapping.
Consequences of this fact will be discussed within the spaces of volume
preserving bijections (i.e. automorphisms) and homeomorphisms of the cube.
We will state the Measure Preserving Lusin Theorem and show (not very
precisely) that ergodicity is generic for volume preserving homeomorphisms.
The talk is based on the book "Typical dynamics of volume preserving
homeomorphisms" by Alpern and Prasad.

24.04.2020

Łukasz Pawelec (SGH)

Denjoy examples and their dimension

We will talk about the Cantor sets occurring when a homeomorphism of the
circle is not conjugated to a rotation (only semiconjugated). I will give
some results on the dimensions and the measures of those sets. Perhaps not
surprisingly these are related to both the diophantine properties of the
rotation number and the Holder exponent of the homeomorphism itself.

4.05.2020 (Monday)

Bohdan Petraszczuk

Rotations, continued fractions and rotational approximation

We will work on a problem: How often and how closely does an orbit return to a neighborhood of its initial point? We will look at a rotation on the unit circle by an irrational number to get the infinitely many of "close return times". After a proof of a theorem which gives us the important information about a sequence of close return times we will study continued fraction algorithm and finally we will be able to interpret the continued fraction in term of close returns.

15.05.2020

James Waterman (Open University)

Wiman–Valiron discs and the Hausdorff dimension of Julia sets of
meromorphic functions

The Hausdorff dimension of the Julia set of transcendental entire and
meromorphic functions has been widely studied. We review results concerning
the Hausdorff dimension of these sets starting with those of Baker in 1975
and continuing to recent work of Bishop. In particular, Baranski, Karpinska,
and Zdunik proved that the Hausdorff dimension of the set of points of
bounded orbit in the Julia set of a meromorphic function with a particular
type of domain called a logarithmic tract is greater than one. We discuss
generalizing this result to meromorphic maps with a simply connected direct
tract and certain restrictions on the singular values of these maps. In
order to accomplish this, we develop tools from Wiman–Valiron theory,
showing that some tracts contain a dramatically larger disk about maximum
modulus points than previously known.

22.05.2020

Mateusz Dembny

Solenoid

In this talk I will present an example of hyperbolic attractor, called the solenoid. First, I will introduce notions such as attracting set, attractor and hyperbolic attractor. Then I will formulate basic theorems about this objects. The solenoid attractor can be given by a specific map. We will study the properties of this map and use this properties in construction of this set. Geometrically, the map can be described as stretching the solid torus out to be twice as long in one direction and wrapping it twice around itself. As a limit of that process we will obtain the solenoid.

29.05.2020

Klaudiusz Czudek (IMPAN)

Towards the solution of some fundamental questions concerning group actions on the circle and codimensionone foliations

There is a famous problem going back to Ghys and Sullivan whether every
minimal action by a group of $C^2$ circle diffeomorphisms is necessarily
ergodic with respect to the Lebesgue measure. I shall present what has
already been proved towards its solution.

5.06.2020

Adam Śpiewak

Dimension of stationary measures with infinite entropy

I will present results of the paper 'Dimension of Gibbs measures
with infinite entropy' by Felipe Pérez, contributing to the study of
geometric properties of stationary measures for infinite iterated function
systems in the infinite entropy case. The main result states that, under
certain assumptions, Gibbs measure of infinite entropy for a Gausslike map

is symbolic exact dimensional (with an explicit formula for the value of
the dimension),

has upper local dimension almost surely equal to the symbolic dimension,

has lower local dimension almost surely equal to zero.
This is in contrast with the finite entropy case, for which all three of the
above notions of dimension coincide.
