Probability distributions on fractal curves of spiderweb type

Abstract: In the talk, we consider a complex-valued random variable \[ \tau = \sum_{n=1}^\infty \frac{2 \varepsilon_{\tau_n}}{3^n}, \] where $(\tau_n)$ is a sequence of independent random variables taking the values $0$, $1$, …, $6$ with the probabilities $p_{0n}$, $p_{1n}$, …, $p_{6n}$, respectively, and $\varepsilon_0$, $\varepsilon_1$, …, $\varepsilon_5$ are the $6$th roots of unity, $\varepsilon_6 = 0$. Structural, spectral, topological, metric, and fractal properties of distribution of this random variable are studied.

We prove that the set of values of the random variable $\tau$ is a self-similar fractal curve of spiderweb type with dimension $\log_3 7$. Its outline is the Koch snowflake.

The Lebesgue structure of distribution of $\tau$ is also studied in detail.

In the case of identically distributed random variables $\tau_n$ we establish that the spectrum of distribution of $\tau$ is a self-similar fractal and essential support of density is a fractal set of Besicovitch–Eggleston type.

Ukrainianclassical analysis and ODEsnumber 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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