On a generalized dimension of self-affine fractals

✍ Scribed by Xing-Gang He; Ka-Sing Lau

• Publisher: John Wiley and Sons
• Year: 2008
• Tongue: English
• Weight: 617 KB
• Volume: 281
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200510666

✦ Synopsis

Abstract

For a d ×d expanding matrix A, we de.ne a pseudo‐norm w (x) in terms of A and use this pseudo‐norm (instead of the Euclidean norm) to define the Hausdorff measure and the Hausdorff dimension dim^w^ ~H~ E for subsets E in R^d^ . We show that this new approach gives convenient estimations to the classical Hausdorff dimension dim^w^ ~H~ E, and in the case that the eigenvalues of A have the same modulus, then dim^w^ ~H~ E and dim~H~ E coincide. This setup is particularly useful to study self‐affine sets T generated by ϕj (x) = A^–1^(x +dj), dj ∈ R^d^ , j = 1, …, N. We use it to investigate the fractality of T for the case that {ϕj }^N^ ~j =1~ satisfying the open set condition as well as the cases without the open set condition. We extend some well‐known results in the self‐similar sets to the self‐af.ne sets. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)