An analog of the Gauss–Kuzmin problem for $A_2$-continued fractions

Abstract: In the talk, the $A_2$-continued fraction representation of numbers $[a_1, a_2, \ldots, a_n, \ldots] \in [0.5, 1]$ with alphabet $\{0.5, 1\}$ is considered. For operator $T([a_1, a_2, \ldots, a_n, \ldots]) = [a_2, a_3, \ldots, a_{n+1}, \ldots]$, we study a sequence of functions $f_n(x) = \lambda(T^{-n}([0.5, x]))$, where $x \in [0.5, 1]$ and $\lambda(\cdot)$ is the Lebesgue measure. We prove that the sequence $(f_n(x))$ converges pointwise to some limiting function and analyze the asymptotics of the corresponding convergence.

Ukrainiandynamical systemsfunctional analysisnumber 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