Reconstruction of Anosov flows from infinity

Every pseudo-Anosov flow $\phi$ in a closed $3$-manifold $M$ gives rise to an action of $\pi_1(M)$ on a circle $S^{1}_{\infty}(\phi)$ from infinity \cite{Fen12}, with a pair of invariant \emph{almost} laminations. From certain actions on $S^{1}$ with invariant almost laminations, we reconstruct flows and manifolds realizing these actions, including all orientable transitive pseudo-Anosov flows in closed $3$-manifolds. Our construction provides a geometry model for such flows and manifolds induced from $\mathcal{D} \times \mathcal{D}$, where $\mathcal{D}$ is the Poincar\'e disk with $\partial \mathcal{D}$ identified with $S^{1}_{\infty}(\phi)$. In addition, our result applies to Cannon conjecture under the assumption that certain group-equivariant sphere-filling Peano curve exists, which offers a description of orientable quasigeodesic pseudo-Anosov flows in hyperbolic $3$-manifolds in terms of group actions on $\partial \mathbb{H}^{3} \times \partial \mathbb{H}^{3} \times \partial \mathbb{H}^{3}$.