Assouad dimensions of Moran sets and Cantor-like sets

Assouad dimensions of complementary sets

“…We begin, in Section 2, by introducing our terminology and notation. There we also derive formulas for the (Lower) Assouad dimensions of the associated sets C a , generalizing the formulas found in [15] and [20] for the special case of central Cantor sets. These formulas will be very useful for the proofs given later in the paper.…”

“…In this case, R is smaller than the size of any gap in A m−1 at level less than h. It follows that B(x, R) ∩ A m−1 is contained in the closed interval between two consecutive gaps of level < h and therefore # (B(x, R) ∩ A m−1 ) ≤ c2 m−h . If, also, r ≥ d m , then diam A m ≤ cr, so N r (A m ) ≤ c. Putting these facts together we see that N r (B(x, R) ∩ A) ≤ C2 m−h , while (15) ensures that Å R r…”

“…Similar arguments show that the same formula holds if C{r j } is a central Cantor set. Previously, the Assouad dimension formula ( 6) was obtained for central Cantor sets, using other methods, by Olson et al in [20]; see also Li et al [15].…”

The Assouad dimension of randomly generated fractals

“…Assouad's original motivation was to study embedding problems, a subject where the Assouad dimension is still playing a fundamental rôle, see [Ol, OR, R]. The concept has also found a home in other areas of mathematics, including the theory of quasi-conformal mappings [H, L, MT], and more recently it is gaining substantial attention in the literature on fractal geometry [K,M,O,Fr3,LLMX,FHOR,ORS]. It is also worth noting that, due to its intimate relationship with tangents, it has always been present, although behind the scenes, in the pioneering work of Furstenberg on micro-sets and the related ergodic theory which goes back to the 1960s, see [Fu].…”