On Hausdorff dimension of Julia sets

Abstract: Roughly speaking, Julia set of a polynomial map is the set of points near which the iterations of this map behave chaotically. It is a fractal set, possessing many interesting properties. A natural question is how large a Julia set can be. One of ways to measure the size of a Julia set is using Hausdorff (and other types of) dimension. In the first part of the talk I will give necessary definitions and a brief overview of the topic. In the second part I will present an approach for studying dimensions of Julia sets and describe computer assisted results obtained using it, solving some open questions in this area (joint work with Igors Gorbovickis and Warwick Tucker).

Ukrainiandynamical systemsnumerical analysis

Audience: researchers in the discipline

( slides )

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Семінар з фрактального аналізу / Fractal analysis seminar

Series comments: Weekly research seminar on fractal analysis (online)

Topics:

• theory of fractals (fractal geometry and fractal analysis)
• Hausdorff–Besicovitch dimension, techniques and methods for its calculation and estimation
• functions and transformations preserving fractal (Hausdorff–Besicovitch, entropic, box-counting, packing, etc.) dimension
• sets of metric spaces that are essential for functions, sets, and dynamical systems
• self-similar, self-affine properties of mathematical objects
• systems of encoding for real numbers (numeral systems) and their applications
• metric number theory and metric theory of representations of numbers
• probabilistic number theory and probabilistic theory of representations of numbers
• measures supported on fractals, particularly singular measures and probability distributions
• nowhere monotonic and nowhere differentiable functions, functions with fractal properties
• theory of groups determined by invariants of transformations preserving tails of representation of numbers, digit frequencies, etc.

The talks are mostly in Ukrainian but English is also acceptable

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