Dynamical Systems

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

 Winter semester Date Speaker Title Abstract 13.10.2017 Anna Zdunik Deterministic and random dynamics of the exponential map This talk will be about the dynamics of the map $z \mapsto \lambda \exp(z)$. We all know how this map acts on the plane. However, the dynamical system generated by such a map is very interesting and I will describe/recall some of its properties. We shall also discuss some random dynamical systems arising from this family. The talk will be elementary and no knowledge of the theory of dynamical systems will be required. 20.10.2017 Krzysztof Lech Indecomposable continua in exponential dynamics The talk will consist of a presentation of a paper by Ł. Pawelec and A. Zdunik, in which they calculate the Hausdorff dimension of some indecomposable continua that appear in exponential dynamics. We will sketch the proof a theorem presented in this paper and follow up by discussing presented applications. In particular, we will prove that the set of points for which all iterations $f^n(z)$ have imaginary part between 0 and $\pi$ (where $f$ is the exponential map) has Hausdorff dimension 1. 27.10.2017 Maria José González (University of Cádiz) A limit theorem for random games We will apply the isoperimetric inequality for boolean function to study the behaviour of the iterates of a special kind of functions, acting on random variables, called selectors. Our main motivation comes from game theory. 3.11.2017 Bartłomiej Żak Random interval transformations We consider two increasing diffeomorphisms applied randomly to the unit interval. We investigate two types of dynamics, the first when the endpoints of the interval are repelling, and the second when they are attracting. We investigate the asymptotic behaviour of the system in both cases and construct invariant measures. 10.11.2017 Reza Mohammadpour Bejargafsheh How projections affect the dimension spectrum of fractal measures We have shown for that for $1< q\leq 2$, $q$-dimensions of $\mu$, pointwise dimension and information dimension are preserved under almost every linear transformation and we have also given some counterexamples for $q>2$ and $0\leq q<1$. 17.11.2017 Bartłomiej Żak Random interval transformations (continuation) We consider two increasing diffeomorphisms applied randomly to the unit interval. We investigate two types of dynamics, the first when the endpoints of the interval are repelling, and the second when they are attracting. We investigate the asymptotic behaviour of the system in both cases and construct invariant measures. 24.11.2017 Łukasz Treszczotko Ergodicity and mixing for stationary processes: point processes approach We present a tool for establishing ergodicity and mixing for a wide class of stationary processes which can be represented as integrals of deterministic functions with respect to Levy white noise random measures. 1.12.2017 Rafał Tryniecki Mathematical problems of blood cells dynamics We will consider model of blood cells dynamics provided by Maria Ważewska-Czyżewska and Andrzej Lasota and analyse stationary solutions of this model. Then we will consider reduced model and prove that in healthy organism permament oscillations in number of blood cells are not possible. 8.12.2017 Piotr Miszczak Synchronization properties of random piecewise isometries We will investigate two results regarding synchronization properties in the case of random double rotations on the circle and a lack of it for its higher dimensions analogue, due to A. Gorodetski and V. Kleptsyn. We will explain what a synchronization means and give a proof of synchronization phenomena, at least in one dimension. 15.12.2017 Reza Mohammadpour Bejargafsheh Multifractal analysis of the Birkhoff sums of Saint-Petersburg potential Let $((0,1], T)$ be the doubling map in the unit interval and $\varphi$ be the Saint-Petersburg potential, defined by $\varphi(x)=2^n$ if $x\in (2^{-n-1}, 2^{-n}]$ for all $n\geq 0$. We consider asymptotic properties of the Birkhoff sum $S_n(x)=\varphi(x)+\cdots+\varphi(T^{n-1}(x))$. With respect to the Lebesgue measure, the Saint-Petersburg potential is not integrable and it is known that $\frac{1}{n\log n}S_n(x)$ converges to $\frac{1}{\log 2}$ in probability. We determine the Hausdorff dimension of the level set $\{x: \lim_{n\to\infty}S_n(x)/n=\alpha\} \ (\alpha>0)$, as well as that of the set $\{x: \lim_{n\to\infty}S_n(x)/\Psi(n)=\alpha\} \ (\alpha>0)$, when $\Psi(n)=n\log n$, $n^a$ or $2^{n^\gamma}$ for $a>1$, $\gamma>0$. 12.01.2018 Jan Kwapisz Box counting dimension of the flexed Sierpiński gasket with non-equal division We calculate the box counting dimension for the flexed Sierpiński gasket, where in each iteration the division of the sides of triangles is unequal. This work is an extension of the article ”Multifractal analysis on the flexed Sierpiński gasket” by Krzysztof Barański. 19.01.2018 Klaudiusz Czudek On the subadditive ergodic theorem I will present an elementary proof of the Kingman's Subadditive Ergodic Theorem given by A. Avila and J. Bochi. The proof does not use the Birkhoff Ergodic Theorem and therefore yields it as a corollary. Preprint. 26.01.2018 Łukasz Chomienia On the exceptional set for absolute continuity of Bernoulli convolutions We will present a paper by Shmerkin in which it is shown that the set of parameters $\lambda\in(1/2,1)$, such that the corresponding self-similar measure of the IFS of the form $(\lambda x+a_{1}, \lambda x+a_{2},\ldots ,\lambda x+a_{m})$ is singular, is of the Hausdorff dimension zero. In the proof we will use some interesting properties of convolutions of some classes of measures and Hochman's results from dimension theory.
 Summer semester Date Speaker Title Abstract 2.03.2018 Adam Śpiewak Absolute continuity of non-homogeneous self-similar measures During the talk I will present results from a recent preprint by S. Saglietti, P. Shmerkin and B. Solomyak concerning properties of non-homogenous self-similar 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 Dini-continuous, 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 chain-recurrent 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 Self-similar 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 super-exponentially 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 re-entering 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 Pomeau-Manneville 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 fine-tuning 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 non-constant rational map.