  We investigate dynamical systems consisting of a locally compact Hausdorff
space equipped with a partially defined local homeomorphism. Important examples
of such systems include self-covering maps, one-sided shifts of finite type
and, more generally, the boundary-path spaces of directed and topological
graphs. We characterise topological conjugacy of these systems in terms of
isomorphisms of their associated groupoids and C*-algebras. This significantly
generalises recent work of Matsumoto and of the second- and third-named
authors.
