A Class of Random Cantor Sets

Abstract

In this paper we study a class of random Cantor sets. We determine their almost sure Hausdorff, packing, box, and Assouad dimensions. From a topological point of view, we also compute their typical dimensions in the sense of Baire category. For the natural random measures on these random Cantor sets, we consider their almost sure lower and upper local dimensions. In the end we study the hitting probabilities of a special subclass of these random Cantor sets.2010 Mathematics Subject Classification. 28A80, 37F40, 60D05.

The Second Generalization of the Hausdorff Dimension Theorem for Random Fractals

“…), dim B (. ), respectively) are presented by Chen [25,26]. Henceforth, in this paper, we consider the n-dimensional unit cube I n (n ≥ 1) in the Euclidean space R n with its conventional Euclidean metric and the Lebesgue measure of λ n (.…”

“…as one of four dimesnions: the Hausdorff dimension, the packing dimension [34], the Assouad dimension [35,36] or the box dimension [37]. The following theorem quantifies these key quantities [25,31].…”

“…Now, the result follows from an application of the Fubini's Theorem and letting k → +∞. [25,26]. Then:…”

“…. (iv) Similar to the case for the Hausdorff dimension, let d = n, M k = m and N k = p k * m n (k ≥ 1) as in [25,26]. Then:…”

The Second Generalization of the Hausdorff Dimension Theorem for Random Fractals

“…), dim B (. ), respectively) are presented in [17,18]. Henceforth, in this paper we consider the n-dimensional unit cube I n (n ≥ 1) in the Euclidean space R n with its conventional Euclidean metric and the Lebesgue measure of λ n (.…”