The rigidity conjecture for branching orbits of pseudo-Anosov flows

Let $\Lambda^s$ and $\Lambda^u$ denote the stable and unstable foliations of a pseudo-Anosov flow $\varphi$. A branching orbit is a closed orbit that lies on a leaf of $\Lambda^s$ or $\Lambda^u$ that is part of a cataclysm in the leaf space. Branching-orbit rigidity conjecture. The set of branching orbits is rigid: on a given $3$-manifold, there are at most finitely many possibilities for the isotopy class of the branching orbits and their degeneracy curves. The conjecture asks whether branching orbits provide another rigid orbit type analogous to singular orbits; its resolution is not given in the supplied text.