Self-Similar Sets 7. A Characterization of Self-Similar Fractals with Positive Hausdorff Measure

Equivalence of positive Hausdorff measure and the open set condition for self-conformal sets

“…We prove that the OSC, the strong OSC, and positivity of the s-dimensional Hausdorff measure are equivalent for conformal contractions; this answers a question of R. D. Mauldin. In the self-similar case, when the contractions are linear, this equivalence was proved by Schief (1994), who used a result of Bandt and Graf (1992), but the proofs in these papers do not extend to the nonlinear setting. …”

“…Bandt and Graf [1] gave a very useful characterization of self-similar sets with positive Hausdorff measure in the similarity dimension. Let A * be the set of finite "words" in the alphabet A = {1, .…”

“…Bandt and Graf [1] proved that H s (K) > 0 if and only if there exists ε > 0 such that for distinct u, v in A * , the maps S u and S v are not ε-relatively close. Building on [1], Schief [11] proved that H s (K) > 0 is equivalent to the OSC and also to the strong OSC.…”

“…Building on [1], Schief [11] proved that H s (K) > 0 is equivalent to the OSC and also to the strong OSC.…”

“…Perhaps surprisingly, the existing proofs of these implications in the self-similar case do not extend to the nonlinear setting. The elegant method of Bandt and Graf [1] for the proof of (b) ⇐⇒ (c) is very much dependent on the set K being precisely self-similar. In several places of [1] it was crucial that j |S j (x)| s = 1 for all x.…”