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

Sylvain Crovisier, Dawei Yang

TL;DR

This work develops a robust framework for topological stability and contraction in flows with index-1 singular dominated splittings, without requiring full hyperbolicity. By leveraging a nonlinear Poincaré flow, a local fibered flow model, identifications between nearby fibers, and a one-dimensional center-stable direction, the authors construct 2D topologically stable discs for μ-a.e. points under mild hypotheses on periodic orbits and toral dynamics. The approach combines plaque theory, δ-interval dynamics, and a lifted dominated-structure analysis to obtain measure-stability results and a Palis-density–type contraction conclusion for compact invariant sets. The results advance understanding of how stable manifolds can persist in singular, non-uniformly hyperbolic settings and pave the way for applications to three-dimensional vector fields and related conjectures.

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 $μ$, we prove that if any periodic orbit within the support of $μ$ (when it exists) has at least one negative Lyapunov exponent, and if the dynamics on the support of $μ$ is not topologically equivalent to an irrational flow on a $2$-torus, then $μ$-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 $μ$. We also establish an analogous conclusion for compact invariant sets $Λ$ with a singular dominated splitting, assuming some mild contraction property (any regular ergodic measure properly supported in $Λ$ 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.

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

TL;DR

This work develops a robust framework for topological stability and contraction in flows with index-1 singular dominated splittings, without requiring full hyperbolicity. By leveraging a nonlinear Poincaré flow, a local fibered flow model, identifications between nearby fibers, and a one-dimensional center-stable direction, the authors construct 2D topologically stable discs for μ-a.e. points under mild hypotheses on periodic orbits and toral dynamics. The approach combines plaque theory, δ-interval dynamics, and a lifted dominated-structure analysis to obtain measure-stability results and a Palis-density–type contraction conclusion for compact invariant sets. The results advance understanding of how stable manifolds can persist in singular, non-uniformly hyperbolic settings and pave the way for applications to three-dimensional vector fields and related conjectures.

Abstract

For vector fields, we study regular ergodic measures whose supports admit singular dominated splittings with one of the bundles having dimension . For such a measure , we prove that if any periodic orbit within the support of (when it exists) has at least one negative Lyapunov exponent, and if the dynamics on the support of is not topologically equivalent to an irrational flow on a -torus, then -almost every point admits a -dimensional topologically stable manifold : we mean that is an embedded disc such that the orbit any point within it converges to the orbit of up to a time-reparametrization. Note that we do not assume any hyperbolicity for . We also establish an analogous conclusion for compact invariant sets with a singular dominated splitting, assuming some mild contraction property (any regular ergodic measure properly supported in 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.
Paper Structure (56 sections, 51 theorems, 83 equations, 4 figures)

This paper contains 56 sections, 51 theorems, 83 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Fibered dynamics with a dominated splitting
  3. Local fibered flow and identifications
  4. Definition of local fibered flows
  5. Definition of identifications
  6. Properties of identifications
  7. Dominated splitting of a local fibered flow
  8. Definition of a dominated splitting
  9. Normally expanded irrational tori
  10. Contraction on periodic orbits and criterion for $2$-domination
  11. Plaque families
  12. Standing assumptions
  13. Plaque family for fibered flows
  14. Plaque family for sequences of diffeomorphisms
  15. Uniqueness
  16. ...and 41 more sections

Key Result

Theorem A

Let $M$ be a compact Riemannian manifold without boundary, $X$ be a $C^2$ vector field over $M$ and $\mu$ be a regular ergodic measure of the flow generated by $X$. Let us assume that on the support of $\mu$: Then $\mu$-almost every point $x$ admits a $2$-dimensional topologically stable disc $V^s(x)$ which is tangent to ${\cal N}^{cs}(x)\oplus X(x)$.

Figures (4)

  • Figure 1: Non-shrinking
  • Figure 2: Shifting to the two sides
  • Figure 3: Aperiodicity
  • Figure 4: Wandering rectangles

Theorems & Definitions (144)

  • Definition 1.1
  • Definition 1.2
  • Theorem A
  • Corollary B
  • Theorem C
  • Definition 2.1: Local fibered flow
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.5: No shear inside orbits
  • Lemma 2.6: Closing lemma
  • ...and 134 more