Exact packing measure of linear Cantor sets

✍ Scribed by De–Jun Feng

• Publisher: John Wiley and Sons
• Year: 2003
• Tongue: English
• Weight: 163 KB
• Volume: 248-249
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200310006

✦ Synopsis

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.