Abstract
In this paper we find a critical condition for nonempty intersection of a limsup random fractal and an independent fractal percolation set defined on the boundary of a spherically symmetric tree. We then use a codimension argument to derive a formula for the Hausdorff dimension of limsup random fractals.
Highlights
A limsup random fractal on RN can be constructed as follows: (i) for each n ≥ 1, let Dn denote the collection of all N -dimensional dyadic hyper-cubes of the form [k12−n, (k1 + 1)2−n] × · · · × [kN 2−n,2−n], where k ∈ ZN+ is an N -dimensional positive integer; (ii) for each n ≥ 1, let {Zn(I) : I ∈ Dn} denote a collection of Bernoulli random variables with distribution P(Zn = 1) = qn; (iii) a limsup random fractal, denoted by A, is defined by
The fast points of Brownian motion considered by Orey and Taylor [16], the thick points of Brownian motion investigated by Dembo, Peres, Rosen, and Zeitouni [1], and the Dvoretzky covering set on the unit circle studied by Li, Shieh and Xiao [10], to name only a few
Under the assumption that the Bernoulli random variables Zn’s are independent, we have succeeded in obtaining a formula for the Hausdorff dimension of limsup random fractals defined on the boundary of a spherically symmetric tree, where a tree is said to be spherically symmetric if and only if all the vertices at the same generation have same number of children
Summary
Under the assumption that the Bernoulli random variables Zn’s are independent, we have succeeded in obtaining a formula for the Hausdorff dimension of limsup random fractals defined on the boundary of a spherically symmetric tree, where a tree is said to be spherically symmetric if and only if all the vertices at the same generation have same number of children The boundaries of these trees include many sets whose Hausdorff and packing dimension are different. As a result of the critical condition derived in Theorem 5.6, we obtain the packing dimension of a fractal percolation set defined on a spherically symmetric tree in Corollary 5.7. We give an example in which we explicitly calculate the dimension of the two random fractals
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