  The Nielsen-Thurston theory of surface diffeomorphisms shows that useful
dynamical information can be obtained about a surface diffeomorphism from a
finite collection of periodic orbits.In this paper, we extend these results to
homoclinic and heteroclinic orbits of saddle points. These orbits are most
readily computed and studied as intersections of unstable and stable manifolds
comprising homoclinic or heteroclinic tangles in the surface. We show how to
compute a map of a one-dimensional space similar to a train-track which
represents the isotopy-stable dynamics of the surface diffeomorphism relative
to a tangle. All orbits of this one-dimensional representative are globally
shadowed by orbits of the surface diffeomorphism, and periodic, homoclinic and
heteroclinic orbits of the one-dimensional representative are shadowed by
similar orbits in the surface.By constructing suitable surface diffeomorphisms,
we prove that these results are optimal in the sense that the topological
entropy of the one-dimensional representative is the greatest lower bound for
the entropies of diffeomorphisms in the isotopy class.
