### THE FINE FRACTAL STRUCTURE OF POINT SETS: AN EXAMPLE OF RIGOROUS ANALYSIS

G.M. Molchan

Abstract

There have been attempts to investigate the multifractal nature of physical objects like earthquake epicenters, star clusters, etc. This is modeled here using a rigorous mathematical analysis of the fine fractal structure of zeroes $Z$ of Brownian motion $w(t)$, $t>0$. We use a natural measure of local time of $w(t)$ on $Z$ and demonstrate that it is a multifractal set with a linear spectral function in the range [1/2, 3/4]. The choice of a measure on a fractal is a procedure that depends on the author's judgement. For this reason we consider $Z$ as the limit of the images $Z_\varepsilon$ with varying degrees of point resolution $\varepsilon$, $\varepsilon\downarrow 0$. The elements of $Z_\varepsilon$ are intervals ($\varepsilon$-clusters) containing points with interpoint distances less than $\varepsilon$. It is shown that the number of $\varepsilon$-clusters of diameter $\varepsilon^\alpha$ ($\alpha$-type) grows like $\varepsilon^{-f(\alpha)}$, where $f(\alpha)$ is a linear function in the interval [1, 2]. The object $Z$ is interesting in that $\varepsilon$-clusters of $\alpha$-type have unexpected limits as $\varepsilon\downarrow 0$. The correct result is obtained from upper (unobservable) limits of $Z_{\varepsilon_n},\:\varepsilon_n=c^{-n},\,c>1,\,n=1,2,\ldots$ or lower limits of $Z_{\varepsilon_n}$ for ultrafast (practically unrealistic) decrease of $\varepsilon_n:\,\varepsilon_n/\varepsilon_{n+1}\to\infty$.

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Computational Seismology, Vol. 3.