The cohomological equation over dynamical systems arising from Delone sets

Abstract

The hull Omega 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 alpha be a continuous G-cocycle over the dynamical system (Omega, R(d)). In this paper we study conditions under which the cohomological equation alpha(omega, x) = psi(omega - x) - psi 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 Omega. For linearly repetitive Delone sets we give a necessary and sufficient condition for solving the cohomological equation in the class of transversally Holder G-cocycles.