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Atomic surfaces, tilings and coincidence I. irreducible case

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Abstract

An irreducible Pisot substitution defines a graph-directed iterated function system. The invariant sets of this iterated function system are called the atomic surfaces. In this paper, a new tiling of atomic surfaces, which contains Thurston’sβ-tiling as a subclass, is constructed. Related tiling and dynamical properties are studied. Based on the coincidence condition defined by Dekking [Dek], we introduce thesuper-coincidence condition. It is shown that the super-coincidence condition governs the tiling and dynamical properties of atomic surfaces. We conjecture that every Pisot substitution satisfies the super-coincidence condition.

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Authors and Affiliations

  1. Mathematical Department of Kanazawa University, Kanazawa, Japan

    Shunji Ito

  2. Mathematical Department of Tsinghua University, 100084, Beijing, China

    H. Rao

  3. Tsuda College, Kodaira, Tokyo, Japan

    H. Rao

Authors
  1. Shunji Ito
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  2. H. Rao
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Correspondence to Shunji Ito or H. Rao.

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The second author is supported by a JSPS Postdoc Fellowship.

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Ito, S., Rao, H. Atomic surfaces, tilings and coincidence I. irreducible case. Isr. J. Math. 153, 129–155 (2006). https://doi.org/10.1007/BF02771781

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  • DOI: https://doi.org/10.1007/BF02771781

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