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 11.10.2019 Artem Dudko (IMPAN) Finding order within the chaos, or why Julia sets are so beautiful Roughly speaking, Julia set of a rational function is the set of points on a complex plane near which the behavior of the iterates of the function is chaotic (unstable, unpredictable, etc.). Despite being related to chaos, it has lots of interesting structure in it. In this talk I will explain in detail what Julia sets are, what kind of structure do they have, and why. 18.10.2019 Matthew Jacques (The Open University) Composition sequences of Möbius maps Let $\mathcal F$ be a finite set of Möbius transformations, each of which maps the unit disc $\mathbb D$ within, but not onto itself. We shall consider sequences generated by repeatedly composing maps drawn from $\mathcal F$. Each such sequence converges locally uniformly in $\mathbb D$ to a constant function. We give a condition that governs whether or not this convergence is uniform. We associate $\mathcal F$ with a directed graph, and use this graph to track the behavior of sequences chosen from $\mathcal F$. In particular, we determine the Hausdorff dimension of the collection of those sequences that converge locally uniformly, but not uniformly on $\mathbb D$. 25.10.2019 Adam Abrams (IMPAN) Introduction to dynamics Dynamical systems studies changes to systems over time. With such a broad description, almost anything can be presented as a dynamical system, and indeed mathematical dynamics relates a wide array of topics. Fortunately, there are some important definitions and results that can be made accessible to a wide range of students. This talk will cover several important notions in dynamics, such as minimality, mixing, and ergodicity. We will address these concepts mainly through instructive examples rather than through detailed proofs. 8.11.2019 Zofia Grochulska Hausdorff and Fourier dimension I will define Hausdorff measure and dimension and describe its basic properties. Methods of computing Hausdorff dimension using energy-like integrals and Fourier transform will be presented together with the notion of Fourier dimension. 15.11.2019 Anna Kryska On Schmidt games and winning sets Following the paper by W. M. Schmidt, we will introduce the Schmidt game. Definition of a winning set associated to it and some of its interesting properties, like maximal Hausdorff dimension and invariance under countable intersections with other winning sets will be discussed. 22.11.2019 Anna Kryska On Schmidt games and winning sets continuation Bohdan Petraszczuk Denjoy's Theorem Our goal will be to prove the Denjoy Theorem. Firstly I will define rotation number of a homeomorphism of the circle and nonlinearity of a $C^1$ real-valued function. In the second part of the talk I will show "Denjoy Counterexamples" which will show that the hypothesis of $C^2$-smoothness is essential. 29.11.2019 Bohdan Petraszczuk Denjoy's Theorem continuation 6.12.2019 Bohdan Petraszczuk Denjoy's Theorem continuation Piotr Rutkowski Dimension of the Julia sets of exponential and sine maps We show that entire functions from the sine family may have Julia of positive area which are not the whole Riemann sphere. Later, our goal is to prove that the Julia set of each member of the exponential family of entire functions has Hausdorff dimension two. 13.12.2019 Piotr Rutkowski Dimension of the Julia sets of exponential and sine maps continuation 20.12.2019 Krzysztof Lech Non-autonomous iteration of quadratic polynomials For a sequence $c_n$, consider the compositions of functions $f_{c_n} = z^2 + c_n$. The Julia and Fatou sets can be defined for such a family of compositions in the same way as in the autonomous case. We shall give a brief overview of results in this setting, and note how it differs from the autonomous case. Then we will answer a question posed by Brück, whether the Julia set for a sequence $c_n$ chosen randomly from a big enough disk is almost always totally disconnected. This is work in progress with Anna Zdunik. 10.01.2020 Adam Śpiewak Probabilistic Takens embedding theorem Takens-type theorems deal with with the problem of reconstructing a dynamical system from a sequence of measurements performed via a one-dimensional observable. They can be also seen as theorems guaranteeing existence of certain embeddings of finite-dimensional dynamical systems into Euclidean spaces. I will present a survey on these results, including a recent probabilistic versions obtained jointly with Krzysztof Barański and Jonatan Gutman. 17.01.2020 Mateusz Dembny Existence of hairs for Zorich maps It is known that for the exponential map $\lambda e^z$, the Julia set consists of uncountably many pairwise disjoint curves tending to infinity. These curves are special in the sense that single curve consists of points sharing "same" iteration under action of $\lambda\exp$. We will show that these have three-dimensional analogues when the exponential function is replaced by a so-called Zorich map. 24.01.2020 Krzysztof Lech Julia sets of random polynomials – part 2 We will continue with discussing conditions for a random Julia set for $z^2 + c_n$ to be totally disconnected. In particular, we will see that one can randomize parameters from the main cardioid of the Mandelbrot set, and still almost always get a totally disconnected Julia set.

 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, ones-density 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 coordinate-wise 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 Kułaga-Przymus. 13.03.2020 Mareike Wolff (Christian-Albrechts-Universität zu Kiel) Exponential polynomials with Fatou and non-escaping 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 so-called 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 non-autonomous 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 non-zero initial point in $\mathbb R^2$ tends to infinity exponentially fast. Later, we will translate this statement into the set-up of Moebius transformations on the upper half-plane. 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 semi-conjugated). 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 codimension-one 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 Gauss-like 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.