Reconstruction of Anosov flows from infinity

Every pseudo-Anosov flow φ in a closed $3$-manifold M gives rise to an action of π₁(M) on a circle S¹∞(φ) from infinity {Fen12}, with a pair of invariant almost laminations. From certain actions on S¹ with invariant almost laminations, we reconstruct flows and manifolds realizing these actions, including all orientable transitive pseudo-Anosov flows in closed $3$-manifolds. Denoting the Poincar\'e disk by D, our construction provides a geometry model for such flows φ and manifolds M by identifying the universal coverings of M with certain spaces induced from D × D and identifying the action of π₁(M) on S¹∞(φ) with an action on ∂ D.