Holomorphic Dynamics

prof. dr hab. Feliks Przytycki, dr Artem Dudko, dr David Martí-Pete

Monday, 14:45-16:15. IMPAN room 408.

 

Programme

  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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).

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