## Abstract

We consider self-affine tilings in ℝ^{n} with expansion matrix φ and address the question which matrices φ can arise this way. In one dimension, λ is an expansion factor of a self-affine tiling if and only if {pipe}λ{pipe} is a Perron number, by a result of Lind. In two dimensions, when φ is a similarity, we can speak of a complex expansion factor, and there is an analogous necessary condition, due to Thurston: if a complex λ is an expansion factor of a self-similar tiling, then it is a complex Perron number. We establish a necessary condition for φ to be an expansion matrix for any n, assuming only that φ is diagonalizable over ℂ. We conjecture that this condition on φ is also sufficient for the existence of a self-affine tiling.

Original language | English |
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Pages (from-to) | 577-593 |

Number of pages | 17 |

Journal | Discrete and Computational Geometry |

Volume | 43 |

Issue number | 3 |

DOIs | |

State | Published - Apr 2010 |

Externally published | Yes |

### Bibliographical note

Funding Information:The work of Kenyon was supported in part by NSERC and NSF. The research of Solomyak was supported in part by NSF grants DMS-0355187 and DMS-0654408.

## Keywords

- Expansion map
- Perron number
- Self-affine tiling