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On a generalized dimension of self‐affine fractals
Authors:Xing‐Gang He  Ka‐Sing Lau
Institution:Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
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 dimw H E for subsets E in R d . We show that this new approach gives convenient estimations to the classical Hausdorff dimension dimw H E, and in the case that the eigenvalues of A have the same modulus, then dimw H E and dimH E coincide. This setup is particularly useful to study self‐affine sets T generated by ?j (x) = A–1(x +dj), dj ∈ R d , j = 1, …, N. We use it to investigate the fractality of T for the case that {?j }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)
Keywords:Box dimension  Hausdorff dimension  iterated function system  open set condition  pseudo norm  self‐affine sets
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