Continuous structurally fractal functions defined in terms of the $Q_s$-representation and nega-$Q_s$-representation of numbers (qualifying seminar)

Abstract: This is a qualifying seminar organized by the Department of Dynamical Systems and Fractal Analysis for the public presentation and discussion of the Ph.D. student's research results obtained for his Ph.D. degree dissertation in specialty 111, Mathematics. The department must provide a detailed report on the scientific novelty and the theoretical and practical value of the dissertation results.

The dissertation is devoted to the applications of finite-symbol representations of real numbers with zero redundancy, such as the $Q_s$-representation and nega-$Q_s$-representation, in the constructive theory of continuous locally complicated functions with structurally fractal properties defined on the interval $[0, 1]$.

The main objects of study are continuous functions preserving the tails of the nega-$Q_s$-representation, nowhere monotonic and nowhere differentiable functions defined in terms of the $Q_s$-representation, and functions related to infinite Bernoulli convolutions governed by the negabinary series.

In the talk, the following main results will be presented:

1. We proved that the set of continuous transformations of the unit interval (i.e., bijective mappings of an interval onto itself) that preserve the tails of the negabinary representation of numbers forms a noncommutative group under the composition (superposition) of transformations with a continuum subgroup of increasing transformations. These transformations are constructed using left- and right-shift operators on the digits of a representation of numbers.
2. The conditions for an infinite Bernoulli convolution governed by the negabinary series to have a uniform, pure discrete, absolutely continuous, singular, or exponential distribution are found.
3. The structure and spectral properties of a random variable whose digits of the negabinary representation form a homogeneous Markov chain are studied.
4. We proved that the nega-$Q_s$-representation of numbers is a re-encoding of the known $Q_s$-representation and generates an identical metric theory. For this system of encoding of numbers, we proved that the group of continuous transformations of the unit interval that preserve the tails of the nega-$Q_s$-representation of numbers is infinite and contains a subgroup of increasing transformations.
5. For a class of continuous locally complicated self-affine functions defined in terms of the $Q_s$-representation of numbers, we computed local Hölder exponents at points with given asymptotic frequencies of digits in their $Q_s$-representation. The conditions for these functions to have continuum level sets are found.
6. For continuous self-affine functions satisfying additional conditions, we described the geometric structure of sets of maximum points and, in particular, showed that these sets can be fractal.

Ukrainianclassical analysis and ODEsfunctional analysisnumber theory

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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