Pisot family self-affine tilings, discrete spectrum, and the meyer property

Jeong Yup Lee, Boris Solomyak

Research output: Contribution to journalArticlepeer-review

21 Scopus citations


We consider self-affne tilings in the Euclidean space and the as-sociated tiling dynamical systems, namely, the translation action on the orbit closure of the given tiling. We investigate the spectral properties of the sys-tem. It turns out that the presence of the discrete component depends on the algebraic properties of the eigenvalues of the expansion matrix φ for the tiling. Assuming that φ is diagonalizable over C and all its eigenvalues are algebraic conjugates of the same multiplicity, we show that the dynamical system has arelatively dense discrete spectrum if and only if it is not weakly mixing, and if and only if the spectrum of φ is a "Pisot family." Moreover, this is equivalent to the Meyer property of the associated discrete set of "control points" for thetiling.

Original languageEnglish
Pages (from-to)935-959
Number of pages25
JournalDiscrete and Continuous Dynamical Systems
Issue number3
StatePublished - Mar 2012
Externally publishedYes


  • Discrete spectrum
  • Meyer sets.
  • No weakly mixing
  • Pisot family
  • Self-áne tilings


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