Assouad dimensions of Moran sets and Cantor-like sets

Dimensions of subsets of cantor-type sets

Let I = [0,1] be the unit interval on the real line andm> 1 be an integer. Let J = {0,1, . . . , m− 1}. For every point x ∈ I , there is a unique base-m representation x = Σk=1 jkm−k with jk ∈ J except for countable many points. Since countable sets do not interfere with our work, we neglect them here. For each j ∈ J , x ∈ [0,1], and n ∈ N, let τj(x,n) = {k : ik = j, 1 ≤ k ≤ n}, then the limit ...

Classifying Cantor Sets by Their Fractal Dimensions

In this article we study Cantor sets defined by monotone sequences, in the sense of Besicovich and Taylor. We classify these Cantor sets in terms of their h-Hausdorff and h-packing measures, for the family of dimension functions h, and characterize this classification in terms of the underlying sequences.

Cantor sets

This paper deals with questions of how many compact subsets of certain kinds it takes to cover the space ω of irrationals, or certain of its subspaces. In particular, given f ∈ (ω\{0}), we consider compact sets of the form Q i∈ω Bi, where |Bi| = f(i) for all, or for infinitely many, i. We also consider “n-splitting” compact sets, i.e., compact sets K such that for any f ∈ K and i ∈ ω, |{g(i) : ...

Julia Sets and Wild Cantor Sets

There exist uniformly quasiregular maps f : R → R whose Julia sets are wild Cantor sets.

On the Dimension of Deterministic and Random Cantor-like Sets