Winter semester 

Date 
Speaker 
Title 
Abstract 
16.10.2020 
Presentation of the scope of the seminar by the coordinators. Fixing the schedule of talks for the winter semester 


23.10.2020 
Kamil Ryduchowski 
Combinatorics of the circle rotation 
Let $T$ be an irrational rotation of the circle and $z$ be a point of the
circle. We consider the problem of close returns, i.e. how often and how
close the orbit of $z$ approaches $z$. We answer that question by giving
recursive formulas as well as studying the asymptotic behaviour. Afterwards
we investigate the relation between that problem and number theory,
especially the continued fraction algorithm.

30.10.2020 
Piotr Oszer 
Introduction to complex dynamics 
Let $f$ be a holomorphic function on the Riemann sphere. We introduce the notion of the Julia set of $f$ and prove its basic properties. Then we investigate the relation between the Julia set and periodic points.

6.11.2020 
Radosław Opoka 
Invariant measures and the PerronFrobenius operator 
Let $(X,\mathcal{B},\mu)$ be a normalized measure space and $\tau : X
\longrightarrow X$ be a nonsingular transformation. We introduce the
PerronFrobenius operator for $X =[a,b] \subset \mathbb{R}$. Then we find
the relationship between this operator and existence of $\tau$invariant
measures $\nu$, absolutely continuous with respect to the Lebesgue measure.
At the end we derive representation of the PerronFrobenius operator for
piecewise monotonic functions.

13.11.2020 
Krzysztof Lech 
Sullivan's Nowanderingdomain Theorem 
Consider a rational map, and the possible connected Fatou components for it. Sullivan proved that rational maps cannot have wandering domains, thus a classification of periodic Fatou components fully describes the Fatou set. We shall discuss the full proof of Sullivan's Nowanderingdomain theorem. This will necessarily include a small introduction to quasiconformal surgery.

20.11.2020 
Krzysztof Lech 
Sullivan's Nowanderingdomain Theorem – continuation 

27.11.2020 
Paweł Cygan 
Equidistribution for random rotation 
Let $f$ be an irrational rotation of the circle. First we prove that there is only one $f$invariant measure $\mu$ and it is the Lebesgue measure. Next we consider a sequence of independent and identically distributed random variables $\omega_{n}$, such that $P(\omega_{n}=\theta_{1})=\frac{1}{2}=P(\omega_{n}=\theta_{2})$ for some fixed $\theta_{1},\theta_{2}$ $\in \mathbb{R}$. Then we define random rotation $f(z)=e^{2\pi i \omega_{n}}z$ and show that if one of the numbers $\theta_1, \theta_2$ is irrational, then we have almost sure equidistribution of the orbit $\text{orb}(z)$ for any $z \in S^{1}$.

4.12.2020 
Piotr Gruza 
Newton's method 
Let $f$ be a complex polynomial, and let us consider a dynamical system given by iterations of rational function $x\mapsto x\frac{f(x)}{f'(x)}$. At first, we investigate its global behaviour (in numerical analysis it is usually studied from a local perspective). In further part, we deliberate its generalization coming from relaxed Newton's method and limit (continuous) case.

11.12.2020 
Adam Śpiewak 
Furstenberg measures 
Let $A_1, A_2, \ldots$ be a random i.i.d. sequence of $2\times 2$ real matrices with determinant $1$. Furstenberg proved in 1960s that under certain assumptions on the distribution of $A_n$:

18.12.2020 
Tomasz Jabłczyński 
Boshernitzan Theorem 
Let $(X, \mathcal{F}, \mu,d,T)$ be a metric, measure preserving dynamical system, such that
$\alpha$dimensional Hausdorff measure of $X$ is finite for an $\alpha
> 0$. We will show that
\[
\liminf_{n\rightarrow \infty} n^{\frac{1}{\alpha}} d(x,T^{n}(x))<\infty
\]
for $\mu$almost all $x\in X$. We will also show uses of Boshernitzan Theorem in finding lower estimates for the
Hausdorff measure of the Cantor set and the Sierpiński triangle in their respective Hausdorff dimensions.

15.01.2021 
Jeremi Zglinicki 
Line, spiral, dense 
If you pick randomly a line from the complex plane, almost surely its image under the function $f(z)=\exp(\exp(z))$ is dense in $\mathbb C$.

22.01.2021 
Łukasz Pawelec (SGH) 
Shrinking targets (bow and arrows not included) 
We will take a brief visit into the field of shrinking targets. The problem starts be taking a sequence of sets whose measures shrink to zero and asking about the set of points that visit (often? always?) the prescribed sequence. This is meant to be a very elementary talk, I plan to show the basic notions, some typical maps and systems and perhaps a few ideas within the proofs.

29.01.2021 
Krzysztof Barański 
Dimensional properties of Julia set for exponential maps 
We describe some results concerning dimension of the Julia sets and their dynamically defined subsets for complex exponential maps $E_\lambda(z) = \lambda e^z$.

Summer semester 

Date 
Speaker 
Title 
Abstract 
5.03.2021 
Leticia Pardo Simón (IMPAN) 
Hyperbolic entire functions with bounded Fatou components 
This talk is named after a paper by Bergweiler, Fagella and Rempe. In this paper, they show that an invariant Fatou component of a hyperbolic transcendental entire function is a Jordan domain if and only if it contains only finitely many critical points and no asymptotic curves. As a result, they obtain criteria for the boundedness of Fatou components and local connectivity of Julia sets for these maps. In this talk, we shall present the results of this paper and sketch the main ideas of proofs.

12.03.2021 
Kamil Ryduchowski 
Towards the Furstenberg x2x3 conjecture 
Consider the circle $\mathbb{T} = \mathbb{R}/\mathbb{Z}$ and for $n\in \mathbb{N}$ let $M_n: \mathbb{T} \rightarrow \mathbb{T}$ be the multiplication
operator, $M_n(x) = nx$ mod $\mathbb{Z}$. Let $a,b\in \mathbb{N}$ be two positive integers which are not powers of the same integer. We will prove the
Furstenberg theorem stating that in this case there is only one infinite closed subset of $\mathbb{T}$ invariant under both $M_a$ and $M_b$ – namely $\mathbb{T}$ itself. The proof will be split into two parts:
first we shall prove a special case, mainly using combinatorics. The general case will follow by applying some well known results from topology and algebra.

19.03.2021 
Paweł Cygan 
On the Apollonian circle packing 
We will deal with the packing problem of Apollonian circles. Our main goal is to formulate and prove a simple version of a theorem by Kontorovich and Oh. This theorem concerns asymptotic behaviour of numbers of circles with radius greater than $\frac{1}{T}$ while $T\to \infty$.

26.03.2021 
Paweł Cygan 
On the Apollonian circle packing – continuation 

31.03.2021 
Radosław Opoka 
Holomorphic approach to the 3n+1 problem 
Let $n$ be a positive integer. If $n$ is even, divide it by $2$; if $n$ is
odd, replace it by $3n+1$. Are we able to reach the number $1$ after finitely
many repetition of this procedure, regardless of value of the starting
number $n$? This is the socalled $3n+1$problem. We will construct entire
holomorphic function which extends $3n+1$problem to complex plane and then we
will investigate properties of the Fatou set of such function.

9.04.2021 
Rafał Tryniecki 
Conditions on continuity of the Hausdorff measure in linear case 
Let $J_n$ be the limit set generated by a family of affine decreasing functions $S_n = \{f_1, f_2, \dots, f_n, \dots\}$ such that $f_k(b_k) = 0$ and $f_k(b_{k+1}) = 1$, where $b_k$ is a sequence decreasing to $0$ such that $b_0 = 1$. We will show that if 1. $\sum\limits_{k = 1}^{\infty} (b_k  b_{k+1})^{h_n} < \infty$, 2. $\lim\limits_{n \to \infty} n^{1h_n} = 1$, 3. $\sup \limits_{k\in \mathbb{N}} \left\{\frac{b_kb_{k+1}}{b_{k+1}} \right\} < \infty$, then the Hausdorff measure of $J_n$ in its Hausdorff dimension (denoted by $h_n$) is continuous, i.e. $\mathcal H_{h_n}(J_n) \to 1$ as $n \to \infty$. 
16.04.2021 
Piotr Oszer 
Introduction to the hyperbolic metric 
We construct and discuss basic properties of the Poincaré metric in the unit disk and, in general, on hyperbolic surfaces. We begin with some brief introduction to the Riemannian surfaces. Then we look at some examples and prove basic theorems (e.g. the SchwarzPick Lemma).

23.04.2021 
Krzysztof Lech 
Lattès maps 
We shall discuss the socalled Lattès maps. After an introduction
we will focus on recent developments and applications, as well as some
examples. The presentation will be fully based on John Milnor's paper "On
Lattès maps".

30.04.2021 
Klaudiusz Czudek (IMPAN) 
The central limit theorem for random circle rotations 
Let us fix two circle rotations and assign to them a probability distribution. When applying the rotations randomly, a Markov process on the circle arise. It was already proved on one of the previous seminars that the trajectory of this random walk is equidistributed almost surely if one of the angles of rotation is irrational. This time I shall discuss whether the central limit theorem holds. Not surprisingly, existing proofs (due to Anna Zdunik and Bence Borda) rely on the Diophantine properties of the angles of rotation.

14.05.2021 
Tomasz Jabłczyński 
Rotation numbers for random dynamical systems on the circle 
We will define random dynamical systems on the circle. We will define rotation number for such systems and prove its existence using Kingman's Subadditive Ergodic Theorem.

21.05.2021 
Jeremi Zglinicki 
Attracting and repelling points of analytic functions 

28.05.2021 
Piotr Gruza 
Fixed points and conjugations 
Continuing the topic started during the previous seminar, we will look at superattracting and neutral fixed points of analytic functions. We will base our discussion on the book "Complex Dynamics" by L. Carleson and T. W. Gamelin.

11.06.2021 
Joanna Horbaczewska (University of Lodz) 
On the behaviour of integrable functions at infinity 
The main purpose of the talk is to examine dependence
between integrability of a function and its behaviour at infinity.
Although convergence of a series
implies convergence to $0$ of the sequence under the sum, usually
integrability of a function does not imply its convergence to $0$. We examine
conditions under which every integrable function is convergent to
$0$ (both in traditional sense and in density). Considered functions can be
defined on $\mathbb N$ (real sequences), on $\mathbb R$ or on any space with a
infinite $\sigma$finite measure (then we have to define what we understand.
by infinity in that space). Achieved results generalize both known dependencies.
between convergence of series and sequences, and relatively new results of C. P.
Niculescu i F. Popovici on convergence in density of functions.
