On random fractals with infinite branching: Definition, measurability, dimensions

Abstract

We discuss the definition and measurability questions of random fractals with infinite branching, and find, under certain conditions, a formula for the upper and lower Minkowski dimensions. For the case of a random self-similar set we obtain the packing dimension.Résumé. Nous discutons les questions de définition et de la mesurabilité des fractales aléatoires avec ramification infinie, et trouvons sous certaines conditions une formule pour les dimensions de Minkowski supérieure et inférieure. En cas d'ensemble aléatoire auto-similaire nous obtenons la dimension d'entassement.

Porosities of Mandelbrot Percolation

“…for all j ∈ IN, where the characteristic function of a set A is denoted by 1 1 A . Further, for σ, τ ∈ I * , the random variables X σ and X τ are identically distributed (see [4,Proposition 1]) and, if σ ≺ τ and τ ≺ σ, they are independent. Thus, X l , l ∈ IN ∪ {0}, have the same distribution.…”

Exact packing dimension of random self-similar sests

“…This set can be considered as a random selfsimilar set as pointed out in [6,Example 6.1]. If we define τ 1] and continue by recursion, then the zero set of the Brownian bridge can be represented as a random selfsimilar set with reduction ratios T 1 = τ 1 and T 2 = 1 − τ 2 which are identically distributed but dependent random variables with joint distribution density (cf. [6, (6.12)])…”

“…The definitions and properties of Hausdorff and packing measures can be found in the monograph by P. Mattila ([8]). The definition of random recursive construction is adduced below, it can also be found in [1], [5], [6], [9].…”