Quasicrystals and Denjoy homeomorphisms

We assume that the atomic positions in a quasicrystal form a repetitive Del one set with a finite Bravais module. Therefore we investigate the dynamica l system(Omega, tau, R-d)arising from the orbit closure of such a set. Usin g the cut-and-project method we construct a Poincare section for the dynami cal system (Omega, tau, R-d) such that the action of R-d reduces to an acti on of Z(d). We obtain d commuting homeomorphisms phi(1),..., phi(d) on a Ca nter set X. In one dimension we relate (X, cp) to the support of the invari ant measure of a homeomorphism on the circle (Denjoy homeomorphism). In thi s way we see that the K-groups with additional structure of the correspondi ng C*-algebra classify these point sets and equivalences between different repetitive Delone sets are established. The discussion includes point sets with an acceptance domain given by a countable union of intervals or with a fractal atomic surface.