Three-symbol Markov representation of numbers and its applications

Abstract: In this talk, we consider a three-symbol Markov representation of numbers in the interval $[0, 1]$. We study topological and metric properties of cylinders for the Markov representation, in particular a basic metric ratio. The notion of a Markov-normal number is introduced in terms of frequencies of digits in the representation of a number. An inversor of digits of the Markov representation of numbers is considered. We prove that it is a continuous strictly decreasing function on the interval $[0, 1]$. Using the notion of cylindrical derivative and normal properties of a number in its Markov representation we find conditions for the derivative of the function to be equal to zero almost everywhere with respect to the Lebesgue measure, i.e., conditions of singularity of the inversor.

Ukrainianclassical analysis and ODEsnumber theoryprobability

Audience: researchers in the discipline

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