We use product systems of C * -correspondences to introduce twisted C *

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 characterize the topological conjugacy of these systems in terms of isomorphisms of their associated groupoids and C*-algebras. This significantly generalizes recent work of Matsumoto and of the second- and third-named authors.

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We show how to recover a discrete twist over an ample Hausdorff groupoid from a pair consisting of an algebra and what we call a quasi-Cartan subalgebra. We identify precisely which twists arise in this way (namely, those that satisfy the local bisection hypothesis), and we prove that the assignment of twisted Steinberg algebras to such twists and our construction of a twist from a quasi-Cartan pair are mutually inverse. We identify the algebraic pairs that correspond to effective groupoids and to principal groupoids. We also indicate the scope of our results by identifying large classes of twists for which the local bisection hypothesis holds automatically.

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