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

TL;DR

This paper proves the existence of topologically stable 2D manifolds around points in certain invariant measures of $C^2$ vector fields, extending stability concepts beyond hyperbolic cases.

Contribution

It introduces conditions under which points in measures with singular dominated splittings have stable manifolds, without requiring hyperbolicity.

Findings

Existence of 2D topologically stable manifolds for measures with singular dominated splittings.

Conditions relating Lyapunov exponents and topological equivalence to irrational flows.

Application potential to the Palis density conjecture in 3D vector fields.

Abstract

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.