Groupoids associated with endomorphisms

Abstract

To a compact Hausdorff space which covers itself, we associate an r r -discrete locally compact Hausdorff groupoid. Its C ∗ {{\mathbf {C}}^ * } -algebra carries an action of the circle allowing it to be regarded as a crossed product by an endomorphism and as a generalization of the Cuntz algebra O p {O_p} . We consider examples related to coverings of the circle and of a Heisenberg 3 3 -manifold.