Asymptotics of Weil–Petersson Geodesics I: Ending Laminations, Recurrence, and Flows

Abstract

We define an ending lamination for a Weil–Petersson geodesic ray. Despite the lack of a natural visual boundary for the Weil–Petersson metric [Bro2], these ending laminations provide an effective boundary theory that encodes much of its asymptotic CAT(0) geometry. In particular, we prove an ending lamination theorem (Theorem 1.1) for the full-measure set of rays that recur to the thick part, and we show that the association of an ending lamination embeds asymptote classes of recurrent rays into the Gromov-boundary of the curve complex $${\mathcal{C}(S)}$$ . As an application, we establish fundamentals of the topological dynamics of the Weil–Petersson geodesic flow, showing density of closed orbits and topological transitivity.