Distribution of unit mass on one fractal self-similar web-type curve

Abstract

In the article, we study structural, spectral, topological, metric and fractal properties of distribution of complex-valued random variable
$\tau=\sum\nolimits_{n=1}^{\infty}\frac{2\varepsilon_{\tau}}{3^n}\equiv\Delta^g_{\tau_1...\tau_n...}$, where $(\tau_n)$ is a~sequence of independent random variables taking the values $0,1,\cdots,6$ with the probabilities $p_{0n}$, $p_{1n},\cdots,p_{6n}$; $\varepsilon_{6}=0$, $\varepsilon_0$, $\varepsilon_1,\cdots,\varepsilon_5$ are 6th roots of unity.

We prove that the set of values of random variable $\tau$ is self-similar six petal snowflake which is a fractal curve $G$ of spider web type with dimension $\log_37$. Its outline is the Koch snowflake.

We establish that $\tau$ has either a discrete or a singularly continuous distribution with respect to two-dimensional Lebesgue measure. The criterion of discreteness for the distribution is found and its point spectrum (set of atoms) is described. It is proved that the point spectrum is a countable everywhere dense set of values of the random variable $\tau$, which is the tail set of the seven-symbol representation of the points of the curve $G$.

In the case of identical distribution of the random variables $\tau_n$ (namely: $p_{kn}=p_k$) we establish that the spectrum of distribution $\tau$ is a self-similar fractal and that the essential support of density is the fractal Besicovitch-Eggleston type set. The set is defined by terms digits frequencies and has the fractal dimension $\alpha_0(E)=\frac{\ln {p_0^{p_0}\cdots p_6^{p_6}}}{-\ln 7}$ with respect to the Hausdorff-Billingsley $\alpha$-measure. The measure is a probabilistic generalization of the Hausdorff $\alpha$-measure. In this case, the random variables $\tau=\Delta^g_{\tau_1\cdots\tau_n\cdots}$ and $\tau'=\Delta^g_{\tau_1'...\tau_n'...}$ defined by different probability vectors $(p_0,\cdots,p_6)$ and $(p'_0,\cdots,p'_6)$ have mutually orthogonal distributions.