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Hausdorff dimension of limsup random fractals. (English) Zbl 1287.60018

A formula for the Hausdorff dimension of limsup random fractals is derived. Consider the collection \(\mathcal{D}_n\) of all \(N\)-dimensional hyper-cubes having the form \([k^12^{-n}, (k^1+1)2^{-n}] \times \cdots \times [k^N2^{-n}, (k^N+1)2^{-n}]\) for an \(N\)-dimensional positive integer \(k \in \mathbb{Z}_+^N\). Let \(\{ Z_n(I):I \in \mathcal{D}_n\}\) be a collection of Bernoulli r.v.’s such that \(\Pr \left\{ Z_n=1 \right\} = q_n\). A limsup random fractal is \(A:= \bigcap_{n \geq 1} \bigcup_{k \geq n} A_k\), where \(A_k= \bigcup_{I \in \mathcal{D}_k, Z_k (I) = 1} I^\circ\) and \(I^\circ\) is the interior of \(I\).
Under the assumption that the Bernoulli random variables are independent, the author derives the following formula for the Hausdorff dimension of a limsup random fractals defined on the boundary of a spherically symmetric tree (\(T\)) \[ \left\| \dim_H(A) \right\|_{L^\infty \left( P \right)} \, = \, \sup \left\{ s \geq 0 : \delta_s(\partial T) > t \right\} \] for \[ \delta_s(\partial T) := \lim_{n \to \infty} \frac{1}{-n} \log \left( \inf_{\mu \in \mathcal{P}(\partial T)} \iint_{d(x, y) \leq e^{-n}} d(x, y)^{-s} \mu(dx) \mu(dy) \right), \] where \(d\) is the metric of the tree on the boundary and \(\mathcal{P}(\partial T)\) is the collection of all Borel probability measures supported on \(\partial T\).

MSC:

60D05 Geometric probability and stochastic geometry
28A80 Fractals
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