Uncountability (continuum cardinality) of the group of continuous transformations of interval that preserve tails of two-symbol representation of numbers

Abstract: We consider continuous transformations that preserve tails of representations, namely, of classic binary, negabinary, $Q_2$-representation, $G_2$-representation and other. Solutions of some class of equations related to left and right shift operators are given. For arbitrary number from the unit interval, we construct the corresponding continuous transformation preserving tails. This is a crucial point to prove that the group of continuous transformations preserving tails is uncountable.

Ukrainianclassical analysis and ODEsgroup theorynumber theory

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