The multisingular partial hyperbolicity dichotomy for vector fields

A vector field $X$ is called multisingular partially hyperbolic if its chain-recurrence set can be split into finitely many compact invariant sets, each admitting a multisingular partially hyperbolic splitting whose center bundle can be split into one-dimensional dominated bundles for the linear Poincaré flow. Multisingular partial hyperbolicity conjecture. Any vector field can be either accumulated by vector fields with a homoclinic tangency, or accumulated by a multisingular partially hyperbolic vector field. This conjecture is presented as an analogue of the corresponding dichotomy for hyperbolic or partially hyperbolic dynamics and is inspired by the cited work of Crovisier, Sambarino, and Yang. Its resolution is not established in the supplied text.