Topology of planar self-affine tiles with collinear digit set

  • Shigeki Akiyama

    University of Tsukuba, Japan
  • Benoît Loridant

    Montanuniversität Leoben, Austria
  • Jörg Thuswaldner

    Montanuniversität Leoben, Austria
Topology of planar self-affine tiles with collinear digit set cover
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Abstract

We consider the self-affine tiles with collinear digit set defined as follows. Let satisfy and be an integral matrix with characteristic polynomial . Moreover, let for some such that are linearly independent. We are interested in the topological properties of the self-affine tile defined by . Lau and Leung proved that is homeomorphic to a closed disk if and only if . In particular, has no cut point. We prove here that has a cut point if and only if . For , the interior of is disconnected and the closure of each connected component of the interior of is homeomorphic to a closed disk.

Cite this article

Shigeki Akiyama, Benoît Loridant, Jörg Thuswaldner, Topology of planar self-affine tiles with collinear digit set. J. Fractal Geom. 8 (2021), no. 1, pp. 53–93

DOI 10.4171/JFG/98