Topologically stable manifolds for index-$1$ singular dominated splittings

For $C^2$ vector fields, we study regular ergodic measures whose supports admit singular dominated splittings with one of the bundles having dimension $1$. For such a measure $\mu$, we prove that if any periodic orbit within the support of $\mu$ (when it exists) has at least one negative Lyapunov exponent, and if the dynamics on the support of $\mu$ is not topologically equivalent to an irrational flow on a $2$-torus, then $\mu$-almost every point $x$ admits a $2$-dimensional topologically stable manifold $V^s(x)$: we mean that $V^s(x)$ is an embedded disc such that the orbit any point within it converges to the orbit of $x$ up to a time-reparametrization. Note that we do not assume any hyperbolicity for $\mu$. We also establish an analogous conclusion for compact invariant sets $\Lambda$ with a singular dominated splitting, assuming some mild contraction property (any regular ergodic measure properly supported in $\Lambda$ must have at least one negative Lyapunov exponent). This result will be used in our future work on the Palis density conjecture for three-dimensional vector fields.