On attractors of Fibonacci maps

Abstract: In 1990s Henk Bruin, Gerhard Keller, Tomasz Nowicki, and Sebastian van Strien showed that smooth unimodal maps with Fibonacci combinatorics and sufficiently high degree of a critical point have a wild attractor, i.e. their metric and topological attractors do not coincide. However, until recently there were no reasonable estimates on the degree of the critical point needed.

In the talk I will present a constructive approach for studying attractors of maps, which are periodic points of a renormalization. Using this approach and rigorous computer estimates, we show that the Fibonacci map of degree $d = 3.8$ does not have a wild attractor, but that for degree $d = 5.1$ the wild attractor exists.

The talk is based on a joint work with Denis Gaidashev.

Ukrainiandynamical systemsnumerical analysis

Audience: researchers in the discipline

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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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