Title

Topology of Connected Self-Similar Tiles in the Plane With Disconnected Interiors

Document Type

Article

Publication Date

5-14-2005

Publication Title

Topology and its Applications

DOI

10.1016/j.topol.2004.11.009

ISSN

0166-8641

Abstract

We study the topological structure of connected self-similar tiles in R2 defined by injective contractions satisfying the open set condition. We emphasize on tiles each of whose interior consists of either finitely or infinitely many components. In the former case, we show in particular that the closure of some component is a topological disk. In the latter case we show that the closure of each component is a locally connected continuum. We introduce the finite tail and infinite replication properties and show that under these assumptions the closure of each component is a disk. As an application we prove that the closure of each component of the interior of the Lévy dragon is a disk.

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