On some metric problems and structure of sequences produced by the left shift operator for symbols of the $A_2$-continued fraction representation with alphabet $\{0.5, 1\}$

Abstract: In the talk, for a number $[a_1, a_2, \ldots, a_n, \ldots]$ given in terms of the $A_2$-continued fraction expansion with alphabet $\{0.5, 1\}$, we consider the left shift operator \[ T([a_1, a_2, \ldots, a_n, \ldots]) = [a_2, a_3, \ldots, a_{n+1}, \ldots]. \] For numbers with two different $A_2$-representations we use the representation that contains period $(0.5, 1)$.

Let \[ T_n(x) = \underbrace{T(T(\ldots T(x)))}_n. \] We study structure of sequences $T_n(x)$ and type of the distribution corresponding to $T_n(x)$. We consider some metric results for the problem when digits $\xi_1$, $\xi_2$, $\ldots$ of the $A_2$-continued fraction representation of number $[\xi_1, \xi_2, \ldots, \xi_n, \ldots]$ are chosen randomly and independently with probabilities $0.5$, respectively.

Ukrainiandynamical systemsnumber theoryprobability

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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