Dynamical Systems - student/PhD seminar at the University of Warsaw

Dynamical Systems

Seminar for graduate/PhD students at the University of Warsaw and all researches interested in Dynamical Systems

Academic year 2018/19

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 (Christian-Albrechts-Universität 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 non-repelling 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 non-repelling 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 Fatou-Shishikura inequality which bounds the number of non-repelling 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 self-similar 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 Pardo-Simón (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 continuum-wise hyperbolicity. We discuss examples of these systems and characterize the possible dynamical phenomena that can occur on cw-hyperbolic transitive homeomorphisms: either they are topologically hyperbolic, or there exist arbitrarily small dynamical balls containing topological semi-horseshoes, that are periodic sets where the dynamics is semiconjugated to the shift of two symbols. We prove cw-hyperbolicity 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.

Summer semester
Date	Speaker	Title	Abstract
8.03.2019	Krzysztof Lech	Conjugacy and counterexample in random iteration	We shall discuss some results by M. Comerford concerning non-autonomous iteration of polynomials. In particular we will look at an example of how non-autonomous iteration yields a different behaviour than the autonomous case.
15.03.2019	Jacek Rutkowski	Dynamics of systems of random interval diffeomorphisms	We shall present some results concerning so-called step skew product systems. A basic result gives the connection between invariant measures for skew product and their natural extentions.
22.03.2019	Jacek Rutkowski	Dynamics of systems of random interval diffeomorphisms, part II
29.03.2019	Jakub Andruszkiewicz	The dimensions of the McMullen carpet	We will compute the Hausdorff dimension of the McMullen carpet. We will also compute the box counting dimension and show under which conditions those two dimensions are equal.
5.04.2019	Zofia Grochulska	Basic local fixed point theory	The behaviour of analytic functions in the neighbourhood of fixed points will be studied with respect to the linearization theorems. The cases for attracting, repelling and rationally neutral fixed points will be proved and for the irrationally neutral fixed points the statement will be presented.
12.04.2019	Zofia Grochulska	Further remarks on local fixed points	An alternative proof for the rationally neutral case will be presented. Some remarks about irrationally neutral fixed points will be discussed, especially a theorem stating that for some multipliers there is no linearization for any polynomial.
26.04.2019	Rafał Tryniecki	Continuity of the Hausdorff measure	In my talk I will present a proof of the continuity of the Hausdorff measure of the attractor of IFS consisting of affine functions on the interval [0,1].
17.05.2019	Vasiliki Evdoridou (The Open University)	Simply connected wandering domains of entire functions	Let $f$ be a transcendental entire function and $U$ be a connected component of the Fatou set of $f$. If $U$ is not eventually periodic then it is called a wandering domain. Sullivan's celebrated result showed that rational functions have no wandering domains. Contrary to rational functions, transcendental entire functions can have wandering domains (simply or multiply connected), but they are not well understood. In this talk we will start with an introduction to the iteration of transcendental entire functions and give some background on wandering domains. We will then focus on simply connected wandering domains and give two different ways of classifying them. These two classification theorems combined give rise to nine different types of simply connected wandering domains. We will discuss the result which is based on approximation theory and allows us to construct examples of all nine types. Finally, we will give details on how some of the examples have been constructed. This is joint work with A.-M. Benini, N. Fagella, P. Rippon and G. Stallard.
24.05.2019	Krystian Kazaniecki	Peetre’s theorem and the trace operator on von Koch’s snowflake	For a domain $\Omega$ with regular boundary Gagliardo proved that the trace operator from $W^1_1(\Omega)$ is a surjection onto $L^1(\partial \Omega)$. Natural question arises if there is a right inverse operator to the trace operator. The negative answer to this question was given by Peetre. He proved that there is no right inverse to the trace operator for the half-plane. In the first part of the talk I will show a new proof of this fact. This proof uses only Whitney decomposition and geometrical properties of classical Banach spaces. In the second part of the talk I will define a trace operator on $W^1_1(K)$, where $K$ is a von Koch’s snowflake. I will construct the right inverse operator to the trace operator on the von Koch’s snowflake. In order to do that I will characterise the trace space as a Arens-Eels space for a carefully chosen metric and I will use Ciesielski’s isomorphism of Hölder and sequence spaces.
31.05.2019	Jakub Andruszkiewicz	Wymiar Hausdorffa i czasy powrotów w układach dynamicznych
7.06.2019	Adam Śpiewak	Krieger's generator theorem	Krieger's generator theorem states that if the Kolomogorov-Sinai entropy of an ergodic automorphism on a probability space is strictly smaller than $\log k$, then there exists a generating partition with $k$ elements. During the talk I will present a proof of this classical result and discuss its recent generalization by Brandon Seward to actions of arbitrary countable groups in terms of the Rokhlin entropy.
14.06.2019	Krzysztof Lech	Sequences for which the Julia set for $z^2 + c_n$ is totally disconnected	We will take a look at a paper by Gong, Qiu and Li in which among other results they prove that the set of sequences $(c_n)$ for which the Julia set of $z^2 + c_n$ is totally disconnected, is of second Baire category. Some other results on non-autonomous iteration of the quadratic family will also be presented.