A Technique in the Topology of Connected Self-Similar Tiles

Document Type

Article

Publication Date

12-2004

Publication Title

Fractals

DOI

10.1142/S0218348X04002653

ISSN

1793-6543

Abstract

Not much is known about the topological structure of a connected self-similar tile whose interior is disconnected, and even less is understood if the interior consists of infinitely many components. We introduce a technique to show that for a large class of self-similar tiles in ℝ2, the closure of each component of the interior is homeomorphic to a disk. This allows us to prove such a result for the Eisenstein set, the fundamental domain of a well-known quadratic canonical number system, and some other well-known fractal tiles.

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