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The cohomological equation over dynamical systems arising from Delone sets

Published online by Cambridge University Press:  26 May 2010

DANIEL CORONEL
DANIEL CORONEL*
Affiliation:
Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Campus San Joaquín, Avenida Vicuña Mackenna 4860, Santiago, Chile (email: acoronel@mat.puc.cl)
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Abstract

The hull Ω of an aperiodic repetitive Delone set P in ℝd is a compact metric space on which ℝd acts continuously by translation. Let G be ℝm or 𝕋m and α be a continuous G-cocycle over the dynamical system (Ω,ℝd) . In this paper we study conditions under which the cohomological equation α(ω,x)=ψ(ω−x)−ψ(ω) has continuous solutions. We give a sufficient condition for general continuous G-cocycles and a necessary condition for transversally locally constant G-cocycles. These conditions are given in terms of the set of first return vectors associated with a tower system for Ω. For linearly repetitive Delone sets we give a necessary and sufficient condition for solving the cohomological equation in the class of transversally Hölder G-cocycles.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 3 , June 2011 , pp. 807 - 833
Copyright
Copyright © Cambridge University Press 2010

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