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