tailieunhanh - On the Minkowski measurability of self-similar fractals in Rd

The question of Minkowski measurability of fractals is investigated for different situations by various authors, notably by M. Lapidus. In dimension one it is known that the attractor of an IFS consisting of similitudes (and satisfying a certain open set condition) is Minkowski measurable if and only if the IFS is of non-lattice type and it was conjectured that this would be true also in higher dimensions. | Turkish Journal of Mathematics Research Article Turk J Math (2013) 37: 830 – 846 ¨ ITAK ˙ c TUB ⃝ doi: On the Minkowski measurability of self-similar fractals in Rd ˙ 1 , Mehmet S ¨ ˙ 1,∗ Ali DENIZ ¸ ahin KOC ¸ AK1 , Yunus OZDEM IR 1 2 ˙ ¨ Andrei RATIU , Adem Ersin UREYEN 1 Department of Mathematics, Anadolu University, 26470, Eski¸sehir, Turkey 2 Department of Mathematics, Melbourne University, Melbourne, Australia Received: • Accepted: • Published Online: • Printed: Abstract: The question of Minkowski measurability of fractals is investigated for different situations by various authors, notably by M. Lapidus. In dimension one it is known that the attractor of an IFS consisting of similitudes (and satisfying a certain open set condition) is Minkowski measurable if and only if the IFS is of non-lattice type and it was conjectured that this would be true also in higher dimensions. Half of this conjecture was proved by Gatzouras in 2000, who showed that the attractor of an IFS (satisfying the open set condition) is Minkowski measurable if the IFS is of non-lattice type. M. Lapidus and E. Pearse give in their recent work in 2010 a sketch of proof of this conjecture. We give in this work, under certain conditions needed for the application of the Lapidus-Pearse theory, a complete detailed proof of this conjecture, filling in the gaps and resolving the difficulties appearing in their sketch of proof. We also give an alternative proof of Gatzouras’ theorem under the same restrictions and give an explicit formula for the Minkowski content. Key words: Self-similar fractals, Minkowski measurability, tube formulas 1. Introduction Let F = J ∪ φj (F ) =: Φ(F ) ⊂ Rd j=1 be a self-similar fractal, where φj : Rd → Rd are similitudes with scaling ratios 0 D . A simple calculation shows that ζ(s) can be expressed as [9, Theorem ] ζ(s) = 832 1− 1 ∑J s j=1 rj for