On the characterization of expansion maps for self-affine tilings

Richard Kenyon, Boris Solomyak

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21 Scopus citations


We consider self-affine tilings in ℝn 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 languageEnglish
Pages (from-to)577-593
Number of pages17
JournalDiscrete and Computational Geometry
Issue number3
StatePublished - Apr 2010
Externally publishedYes

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.


  • Expansion map
  • Perron number
  • Self-affine tiling


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