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Geometric realization for substitution tilings

Published online by Cambridge University Press:  05 December 2012

MARCY BARGE  and
JEAN-MARC GAMBAUDO
MARCY BARGE
Affiliation:
Montana State University, Bozeman, MT, 59717, USA (email: mbarge@math.montana.edu)
JEAN-MARC GAMBAUDO
Affiliation:
INLN, UMR CNRS 7335, Université de Nice-Sophia Antipolis 1361, route des Lucioles, 06560 Valbonne, France (email: jean-marc.gambaudo@inln.cnrs.fr)
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Abstract

Given an n-dimensional substitution Φ whose associated linear expansion Λ is unimodular and hyperbolic, we use elements of the one-dimensional integer Čech cohomology of the tiling space ΩΦ to construct a finite-to-one semi-conjugacy G:ΩΦ→𝕋D, called a geometric realization, between the substitution induced dynamics and an invariant set of a hyperbolic toral automorphism. If Λ satisfies a Pisot family condition and the rank of the module of generalized return vectors equals the generalized degree of Λ, G is surjective and coincides with the map onto the maximal equicontinuous factor of the ℝn-action on ΩΦ. We are led to formulate a higher-dimensional generalization of the Pisot substitution conjecture: if Λ satisfies the Pisot family condition and the rank of the one-dimensional cohomology of ΩΦ equals the generalized degree of Λ, then the ℝn-action on ΩΦhas pure discrete spectrum.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 2 , April 2014 , pp. 457 - 482
Copyright
©2012 Cambridge University Press 

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