Exact packing measure of linear Cantor sets

Abstract

Let K be the attractor of a linear iterated function system Sj x = ρjx + bj (j = 1, . . . , m) on the real line satisfying the open set condition (where the open set is an interval). It is well known that the packing dimension of K is equal to α, the unique positive solution y of the equation m j=1 ρ y j = 1; and the α-dimensional packing measure P α (K) is finite and positive. Denote by µ the unique self-similar measure for the IFS Sj m j=1 with the probability weight ρ α j m j=1 . In this paper, we prove that P α (K) is equal to the reciprocal of the so-called "minimal centered density" of µ, and this yields an explicit formula of P α (K) in terms of the parameters ρj, bj (j = 1, . . . , m). Our result implies that P α (K) depends continuously on the parameters whenever j ρj < 1.

On the packing measure of the Sierpinski gasket

“…This can be checked by noting that, on the one hand, any ball centered in S 2 and contained in R can be reflected across the altitude h 1 of the triangle T. On the other hand, any ball centered in S 1 and contained in R can be reflected across the altitude h 2 of the triangle T (see figure 2). This yields, by (9), to density equivalent balls centered in S 0 . This concludes the proof of (12) as it shows that any ball centered in S 1 ∪ S 2 and contained in R is density equivalent to another ball centered in S 0 .…”

“…However, as r tends to 1/2 and the gap between the three copies under the similarities in Ψ r goes to zero, the computational time grows at a fatal rate, and the method used in the references mentioned above becomes useless (see section 4.3 for further discussion). In fact linear self-similar sets and the n-dimensional unit cubes are the unique class of self-similar sets satisfying the OSC whose packing measure is known (see [9,10] and [31]).…”

The Sizes of Rearrangements of Cantor Sets

“…As is the case with Hausdorff measures, the sharp value of c and the exact set from C a which minimizes P h 0 is unknown, even in the case of the middle-third Cantor set C a . It is known that 4 s = P s (C a ) = lim sup n(r n /n) s = 1, where s = log 2/ log 3 (see [7,8]).…”