Inflations of self-affine tilings are integral algebraic Perron
We prove that any expanding linear map that is the inflation map in an inflation-substitution process generating a self-affine tiling of is integral algebraic and Perron. This means that is linearly conjugate to a restriction of an integer matrix to a subspace 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 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 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 -action on the tiling space. The subspace is obtained then by ergodic averaging of the cocycle. More broadly, we assemble a powerful framework for studying self-affine tiling spaces.
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