Groupoids Associated with Endomorphisms

To a compact Hausdorff space which covers itself, we associate an r-discrete locally compact Hausdorff groupoid. Its 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 Oₚ. We consider examples related to coverings of the circle and of a Heisenberg $3$-manifold.