A set of quantum states, dynamically related to the classical periodic orbits of a chaotic map, is used as a basis in which the description of the eigenstates of its quantum version is greatly simplified. This set can be improved with the inclusion of short time propagation along the stable and unstable manifolds of the periodic orbits resulting in a construction similar to the scar functions of Vergini [J. Phys. A 33, 4709 (2000)]. The average participation ratio is used to quantify the quality of the basis.