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Thick and thin points for random recursive fractals. (English) Zbl 1032.60048

The class of random recursive fractals provides a wide class of fractal sets which can be used for image generation and to describe models for physical systems with irregular fractal structure. The authors’ main concern is the local structure and properties of the natural measure on the random recursive fractal \(F\). In particular, they concentrate on the thick and thin points and show that the usual multifractal spectrum is trivial by showing that all points in the fractal \(F\) have the same local dimension, i.e., \[ \lim_{r \to 0} \frac{ \log \mu ( B_r(x))}{- \log r} = \alpha \quad ( \forall x \in F), \] where \(\mu\) is the renormalized restriction of Lebesgue measure to \(F\), \(\alpha\) is the Hausdorff dimension of the fractal, and \(B_r(x)\) denotes the ball of radius \(r\) at \(x \in {\mathbb R}^d\). Moreover, it is proved for thin points that there are fluctuations on a log scale which can occur when considering the local dimension: namely, \[ C_0 \leqslant \lim_{r \to 0} \inf_{x \in F} \frac{\mu( B_r(x))}{r^{\alpha}(\log r)^{- 1/ \eta}} \leqslant C_1 \quad P\text{-a.s.} \quad (\exists C_0, C_1 > 0), \] where \(\eta\) is an exponent with an explicit description. Finally, by examining the local behavior of the measure at typical points in the set, the authors establish the size of fine fluctuations in the measure: i.e., \[ C_2 \leqslant \liminf_{r \to 0} \frac{\mu( B_r(x))}{r^{\alpha}(\log \log r)^{-1/ \eta}} \leqslant C_3 \quad P\text{-a.s.} \quad (\exists C_2, C_3 > 0) \] holds for \(\mu\)-a.e. \(x \in F\). Of course, the corresponding results for the thick points are also derived. Furthermore, exact expressions for these constants on the boundary of the tree which describes the fractal are obtained as well. The key point to their results is a large deviation principle (LDP) for general branching processes, which is an extension of LDP for the classical Galton-Watson processes, cf. J. D. Biggins and N. H. Bingham [Adv. Appl. Probab. 25, 757-772 (1993; Zbl 0796.60090)]. The authors use a new renewal theoretic approach, and a general branching process can be used to describe the class of random recursive fractals in question, and besides, for a subclass of these processes, they can establish these large deviation asymptotics. By translating these results to the fractal, they succeed in getting precise estimates on the local structure of the measure.
For related works, see e.g. Q. Liu [Ann. Inst. Henri Poincaré, Probab. Stat. 37, 195-222 (2001; Zbl 0986.60080)] and P. Mörters and N.-R. Shieh [Stat. Probab. Lett. 58, 13-22 (2002; Zbl 1005.60090)].

MSC:

60G57 Random measures
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
28A80 Fractals
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