Inventiones mathematicae

, Volume 205, Issue 1, pp 173–220 | Cite as

Inflations of self-affine tilings are integral algebraic Perron

  • Jarosław Kwapisz


We prove that any expanding linear map \(\phi : \mathbb {R}^{\mathtt {d}} \rightarrow \mathbb {R}^{\mathtt {d}}\) that is the inflation map in an inflation-substitution process generating a self-affine tiling of \(\mathbb {R}^\mathtt {d}\) is integral algebraic and Perron. This means that \(\phi \) is linearly conjugate to a restriction of an integer matrix to a subspace \({\mathcal {E}}\) satisfying a maximal growth condition that generalizes the characterization of Perron numbers as numbers that are larger than the moduli of their algebraic conjugates. The case of diagonalizable \(\phi \) has been previously resolved by Richard Kenyon and Boris Solomyak, and it is rooted in Thurston’s idea of lifting the tiling from the physical space \(\mathbb {R}^\mathtt {d}\) to a higher dimensional mathematical space where the tiles (their control points) sit on a lattice. The main novelty of our approach is in lifting the inflation-substitution process to the mathematical space and constructing a certain vector valued cocycle defined over the translation induced \(\mathbb {R}^\mathtt {d}\)-action on the tiling space. The subspace \({\mathcal {E}}\) is obtained then by ergodic averaging of the cocycle. More broadly, we assemble a powerful framework for studying self-affine tiling spaces.



The author is grateful to Boris Solomyak for encouragement and explanation of the arguments in [10] and acknowledges the support, in the final stages of the preparation of the manuscript, by the Research Expansion Fund under the auspices of the Vice President for Research, MSU.


  1. 1.
    Benedetti, R., Gambaudo, J.-M.: On the dynamics of G-solenoids. Applications to Delone sets. Ergodic Theory Dyn. Syst. 23, 673–691 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Bindi, L., Steinhardt, P.J., Yao, N., Lu, P.J.: Natural quasicrystals. Science 324(5932), 1306–1309 (2009)CrossRefGoogle Scholar
  3. 3.
    Bru, R., Rodman, L., Schneider, H.: Extensions of Jordan bases for invariant subspaces of a matrix. In: Proceedings of the First Conference of the International Linear Algebra Society (Provo, UT, 1989), vol. 150, pp. 209–225 (1991)Google Scholar
  4. 4.
    Friedberg, S.H., Insel, A.J., Spence, L.E.: Linear Algebra, 4th edn. Prentice Hall Inc., Upper Saddle River (2003)zbMATHGoogle Scholar
  5. 5.
    Gohberg, I., Lancaster, P., Rodman, L.: Invariant Subspaces of Matrices with Applications, Classics in Applied Mathematics, vol. 51. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2006). (reprint of the 1986 original)CrossRefGoogle Scholar
  6. 6.
    Kellendonk, J.: Non-commutative geometry of tilings and gap labelling. Rev. Math. Phys. 7, 1133–1180 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Kellendonk, J., Tilings, Putnam I.: \(C^*\)-theory. In: Directions in Mathematical Quasicrystals, CRM Monogr. Ser., vol. 13, pp. 177–206. Amer. Math. Soc., Providence (2000)Google Scholar
  8. 8.
    Kenyon, R.: The construction of self-similar tilings. Geom. Funct. Anal. GAFA 6(3), 471–488 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Kenyon, R.: Self-Similar Tilings. PhD thesis, Princeton Univeristy (1990)Google Scholar
  10. 10.
    Kenyon, R., Solomyak, B.: On the characterization of expansion maps for self-affine tilings. Discrete Comput. Geom. 43(3), 577–593 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Kwapisz, J.: Rigidity and mapping class group for abstract tiling spaces. Ergodic Theory Dyn. Syst. 31(6), 1745–1783 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Lagarias, J.C.: Geometric models for quasicrystals I. Delone sets of finite type. Discrete Comput. Geom. 21(2), 161–191 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Lagarias, J.C., Wang, Y.: Substitution Delone sets. Discrete Comput. Geom. 29(2), 175–209 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Lee, J.-Y., Moody, R.V., Solomyak, B.: Pure point dynamical and diffraction spectra. Ann. Henri Poincar 3(5), 1003–1018 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Lee, J.-Y., Moody, R.V., Solomyak, B.: Consequences of pure point diffraction spectra for multiset substitution systems. Discrete Comput. Geom. 29(4), 525–560 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Lee, J.-Y., Solomyak, B.: Pure point diffractive substitution Delone sets have the Meyer property. Discrete Comput. Geom. 39(1–3), 319–338 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Lind, D.A.: The entropies of topological markov shifts and a related class of algebraic integers. Ergodic Theory Dyn. Syst. 4(6), 283–300 (1984)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Mandelkern, M.: Metrization of the one-point compactification. Proc. Am. Math. Soc. 107(4), 1111–1115 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Manning, A.: A Markov partition that reflects the geometry of a hyperbolic toral automorphism. Trans. Am. Math. Soc. 354(7), 2849–2863 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Moody, R.V.: Model sets: a survey. In: Axel, F., Dénoyer, F., Gazeau, J.-P. (eds.) From Quasicrystals to More Complex Systems, Centre de Physique des Houches, vol. 13, pp. 145–166. Springer, Berlin (2000)CrossRefGoogle Scholar
  21. 21.
    Robinson, E.A., Jr.: Symbolic dynamics and tilings of \({\mathbb{R}}^d\). In: Symbolic Dynamics and Its Applications, Proc. Sympos. Appl. Math., vol. 60, pp. 81–119. Amer. Math. Soc., Providence (2004)Google Scholar
  22. 22.
    Rudolph, D.J.: Markov tilings of \({\bf R}^n\) actions. In: Measure and Measurable Dynamics (Rochester, NY, 1987), Contemp. Math., vol. 94, pp. 271–290. Amer. Math. Soc., Providence (1989)Google Scholar
  23. 23.
    Sinaĭ, Ja. G.: Construction of Markov partitionings. Funkcional. Anal. i Priložen., 2(3), 70–80 (Loose errata) (1968)Google Scholar
  24. 24.
    Solomyak, B.: Nonperiodicity implies unique composition for self-similar translationally finite tilings. Discrete Comput. Geom. 20(2), 265–279 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Solomyak, B.: Dynamics of self-similar tilings. Ergodic Theory Dyn. Syst. 17(3), 695–738 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Thurston, W.: Groups, tilings and finite state automata. Summer 1989 AMS Colloquium Lectures, Research Report GCG 1, Geometry CenterGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Mathematical SciencesMontana State UniversityBozemanUSA

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