The boundaries of self-similar tiles in Rn

✍ Scribed by James Keesling

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 193 KB
• Volume: 94
• Category: Article
• ISSN: 0166-8641
• DOI: 10.1016/s0166-8641(98)00031-5

✦ Synopsis

Let K be a self-similar set in R n which has similarity dimension n and nonempty interior. In this paper it is shown that the topological boundary of K has Hausdorff dimension less than n. Examples are given to show that although the dimension of the boundary is strictly less than n, it may be arbitrarily close to n.

Let K be a self-similar set in any complete metric space X such that K satisfies the strong open sets condition (SOSC). A recent result of A. Schief shows that dim H K = α where α is the similarity dimension of K. If O is the open set given by the SOSC, then it is shown in this paper that dim H (K \O) < α. More generally, if A is any inverse invariant closed subset of K, then dim H A < α.