A topological approach and understanding to the detection of unstable periodic orbits based on a recently proposed method [Phys. Rev. Lett. 78, 4733 (1997)] is developed. This approach provides a classification of the set of transformations necessary for finding the orbits. Applications to the Ikeda and Hénon map are performed, allowing a study of the distributions of Lyapunov exponents for high periods. In particular, the properties of the least unstable orbits up to period 36 are investigated and discussed.