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A Survey on Topological Properties of Tiles Related to Number Systems

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Abstract

In the present paper we give an overview of topological properties of self-affine tiles. After reviewing some basic results on self-affine tiles and their boundary we give criteria for their local connectivity and connectivity. Furthermore, we study the connectivity of the interior of a family of tiles associated to quadratic number systems and give results on their fundamental group. If a self-affine tile tessellates the space the structure of the set of its ‘neighbors’ is discussed.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Niigata University, Ikarashi 2-8050, Niigata, 950-2181, Japan

    Shigeki Akiyama

  2. Department of Mathematics and Statistics, Leoben University, Franz-Josef-Strasse 18, A-8700, Leoben, Austria

    Jörg M. Thuswaldner

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  1. Shigeki Akiyama
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  2. Jörg M. Thuswaldner
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Akiyama, S., Thuswaldner, J.M. A Survey on Topological Properties of Tiles Related to Number Systems. Geom Dedicata 109, 89–105 (2004). https://doi.org/10.1007/s10711-004-1774-7

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