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March 9, 2020, 10:00-11:30. IMPAN, room 408.
Łukasz Pawelec (SGH Warsaw School of Economics)
Hairs and coding for exponential maps
I want to talk about properties of certain set occurring naturally in the dynamics of the exponential map \lambda e^z. I will talk about some topological properties and the Hausdorff dimension of hairs and other more general subsets. This will be done both in the standard case and for the non-autonomous dynamics (\lambda changing at every step).
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February 17, 2019, 10:00-11:30. IMPAN, room 408.
Artem Dudko (IMPAN)
Buff-Chéritat construction of quadratic Julia sets of positive measure - PART II: Key lemma for the Cremer case
In this talk I will sketch the proof of the key lemma for the construction of quadratic Cremer Julia sets of positive measure. Roughly speaking, this lemma states that for an appropriate angle a of bounded type one can find a sequence of angles a_n of bounded type convergent to a so that 1) the corresponding Siegel maps exp(2piia_n)z+z^2 have cycles converging to zero, 2) the corresponding Siegel disks are not much smaller then the Siegel disk of exp(2piia)z+z^2.
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February 3, 2019, 10:00-11:30. IMPAN, room 408.
Artem Dudko (IMPAN)
Buff-Chéritat construction of quadratic Julia sets of positive measure - PART I: Outline of the construction
In this minicourse I will explain the celebrated examples of quadratic Julia sets of positive aree constructed by Xavier Buff and Arnaud Chéritat. These examples are of three types: with Cremer fixed point, with Siegel fixed point, and infinitely satellite renormalizable maps. In the first talk I will present the key ingredients and make an outline of the constructions and the proofs.
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December 9, 2019, 15:00-16:30. IMPAN, room 408.
Artem Dudko (IMPAN)
On Cremer Julia sets
In this talk I will review classical properties of Cremer Julia sets, such as local non-connectivity and existence of hedgehogs, and sketch some proofs.
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November 18, 2019, 15:00-16:30. IMPAN, room 408.
David Martí-Pete (IMPAN)
A transcendental Julia set of dimension 1 - PART III: Other properties and modifications
In the last talk of this series, we will review some of the other properties of the function constructed by Bishop whose Julia set has Hausdorff dimension 1, such as that it has order 0 or that the packing dimension of the Julia set is 1. It is a well-known fact that the packing dimension is greater or equal than the Hausdorff dimension. We will also sketch the modification of this construction by Jack Burkart, to obtain functions whose packing dimension forms a dense subset of the interval (1,2). In these examples, the Hausdorff dimension and the packing dimension can be chosen to be arbitrarily close.
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November 4, 2019, 15:00-16:30. IMPAN, room 408.
Leticia Pardo-Simón (IMPAN)
A transcendental Julia set of dimension 1 - PART II: Main ideas of the proof
This mini-series of seminars aims to study Bishop's example of a transcendental entire function f whose Julia set J(f) has Hausdorff dimension one. In the previous session, historical context of the result, as well as an overview on how the function f is constructed, was given. In this talk, I will provide more details on the dynamics of f, as well as comment on the main ideas of the proof of the result that ensure the Hausdorff dimension of J(f) to be one.
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October 28, 2019, 15:00-16:30. IMPAN, room 408.
David Martí-Pete (IMPAN)
A transcendental Julia set of dimension 1 - PART I: Introduction
Abstract: Baker proved that for transcendental entire functions, the Julia set always has dimension greater than or equal to one. Stallard was able to produce functions in the Eremenko-Lyubich class whose Julia set has any Hausdorff dimension strictly greater than one and less or equal to two. In fact, she also proved that in that class, the Hausdorff dimension of the Julia set is strictly larger than one. Recently, Bishop constructed the first example of a transcendental entire function whose Julia set has Hausdorff dimension equals one. This function is given as an infinite product and has a multiply connected wandering domain. The goal of this seminar series is to give a detailed proof of Bishop's result. In this first talk, we will start by introducing the main definitions and give a rough sketch of the proof.
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May 13, 2019, 14:45-16:15. IMPAN, room 408.
David Martí-Pete (IMPAN)
Dimension in transcendental dynamics - PART III: The Eremenko-Lyubich class
Abstract: The Eremenko-Lyubich class B consists of the transcendental entire functions for which the set of singular values is bounded, and for such maps the escaping set is contained in the Julia set. On 2005, Rippon and Stallard proved that for functions in this class, both the escaping set and the Julia set have packing dimension equal to 2. On 1996, Stallard proved that the Hausdorff dimension of the Julia set is strictly greater than 1 in class B, while on 2010, together with Rempe, they constructed a function for which the escaping set has Hausdorff dimension equal to 1. If we further assume that the function has finite order, then Baranski and Schubert proved independently that the Hausdorff dimension of the escaping set (and hence also the Julia set) equals 2. We will give an overview of what is known about dimension in class B.
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May 6, 2019, 14:45-16:15. IMPAN, room 408.
David Martí-Pete (IMPAN)
Dimension in transcendental dynamics - PART II: The exponential family
Abstract: We will study the exponential family, given by $E_\lambda(z)=\lambda \exp z$, for $\lambda\in\mathbb{C}\setminus\{0\}$. On 1987, McMullen proved that the Julia set (as well as the escaping set) of such functions always has Hausdorff dimension two. For $0<\lambda<1/e$, the Julia set of $E_\lambda$ consists of an uncountable union of curves known as a Cantor bouquet. On 1999, Karpińska proved that the set of endpoints of such curves has Hausdorff dimension two, but the curves without the endpoints have Hausdorff dimension one. Nowadays, this is known as Karpińska's paradox. On 2003, Urbański and Zdunik showed that the set of non-escaping endpoints have Hausdorff dimension less than two, hence concluding that the Hausdorff dimension of the Julia set sits on the set of escaping endpoints for such functions.
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April 29, 2019, 14:45-16:15. IMPAN, room 408.
David Martí-Pete (IMPAN)
Dimension in transcendental dynamics - PART I: Introduction to fractal dimension
Abstract: In this talk, we will review the main notions of fractal dimension, including the packing dimension, the box dimension and the Hausdorff dimension of a set. We will prove their basic properties and give some examples of sets for which it is possible to compute the dimension explicitly.
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April 15, 2019, 16:00-17:30. IMPAN, room 405.
Polina Vytnova (University of Warwick)
The thermodynamic approach to the Hausdorff dimension: old and new
Abstract: In 2001, Jenkinson and Pollicott proposed an efficient algorithm for estimating the Hausdorff dimension of dynamically defined sets based on thermodynamic formalism. The algorithm came with error estimates, but the amount of computer time required to obtain rigourous results was more than the age of the universe. Recently we managed to improve the error bounds, which now allow to obtain rigourous results keeping the computational time under 24 hours. I am planning to discuss the essential ingredients for the method to work in particular settings of the (hyperbolic) Julia sets.
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March 4, 2019, 14:45-16:15. IMPAN, room 408.
Artem Dudko (IMPAN)
On Lebesgue measure of the Feigenbaum Julia set - PART II
Abstract: In this lecture I will briefly remind the key definitions and statements from the first lecture and give a sketch of the proof of the main result: the Julia set of the Feigenbaum map has Hausdorff dimension less than two.
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February 18, 2019, 14:30-16:00. IMPAN, room 408.
Artem Dudko (IMPAN)
On Lebesgue measure of the Feigenbaum Julia set - PART I
Abstract: In this mini-course I will give a detailed exposition of the proof of the following result: the Julia set of the Feigenbaum map has Hausdorff dimension less than two and therefore has zero Lebesgue measure (joint work with S. Sutherland).
