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Inventiones mathematicae

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

Inflations of self-affine tilings are integral algebraic Perron

  • Jarosław Kwapisz
Article
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Abstract

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.

Notes

Acknowledgments

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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Mathematical SciencesMontana State UniversityBozemanUSA

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