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On the Characterization of Expansion Maps for Self-Affine Tilings

On the Characterization of Expansion Maps for Self-Affine Tilings,10.1007/s00454-009-9199-6,Discrete & Computational Geometry,Richard Kenyon,Boris Sol

On the Characterization of Expansion Maps for Self-Affine Tilings   (Citations: 4)
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We consider self-affine tilings inRn with expansion matrixand address the question which matricescan arise this way. In one dimension, � is an expansion factor of a self-affine tiling if and only if|�| is a Perron number, by a result of Lind. In two dimensions, whenis a similarity, we can speak of a complex expansion factor, and there is an analogous necessary condition, due to Thurston: if a complexis an expansion factor of a self-similar tiling, then it is a complex Perron number. We establish a necessary condition forto be an expansion matrix for any n, assuming only thatis diagonalizable over C. We conjecture that this condition onis also sufficient for the existence of a self-affine tiling.
Journal: Discrete & Computational Geometry - DCG , vol. 43, no. 3, pp. 577-593, 2010
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