The cohomological equation over dynamical systems arising from Delone sets

The hull Ω of an aperiodic repetitive Delone set P in ℝ^d is a compact metric space on which ℝ^d acts continuously by translation. Let G be ℝ^m or T^m and α be a continuous G-cocycle over the dynamical system (Ω, ℝ^d). 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.