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On dynamical coherence of partially hyperbolic flows

Mounib Abouanass

TL;DR

The paper addresses the integrability of the subcenter bundle for partially hyperbolic flows and proves that a flow with a flow-invariant subcenter foliation $\mathcal{F}^{\hat{c}}$ that is compact with trivial holonomy is complete, hence dynamically coherent. The authors develop a transverse-foliation framework, introduce unstable and weak unstable projections, and use holonomy control to demonstrate center and subcenter completeness, culminating in a local product structure in the leaf space $M/\mathcal{F}^{\hat{c}}$. They extend discrete results on uniformly compact center foliations to the flow setting, establish global $su\Phi$-holonomy maps under suitable regularity, and derive quasi-shadowing properties in the quotient. The results have implications for transversely holomorphic partially hyperbolic flows on seven-dimensional manifolds and connect continuous-time dynamics with established discrete-time coherence theory. Overall, the work provides structural robustness—completeness, coherence, and local product structure—for partially hyperbolic flows under controlled holonomy and compactness assumptions.

Abstract

In this paper, we introduce the notion of dynamical coherence for a partially hyperbolic flow $(\varphi^t)$ on a smooth compact manifold $M$, and prove it under the assumption that there exists a compact foliation with trivial holonomy which integrates the subcenter distribution.

On dynamical coherence of partially hyperbolic flows

TL;DR

The paper addresses the integrability of the subcenter bundle for partially hyperbolic flows and proves that a flow with a flow-invariant subcenter foliation that is compact with trivial holonomy is complete, hence dynamically coherent. The authors develop a transverse-foliation framework, introduce unstable and weak unstable projections, and use holonomy control to demonstrate center and subcenter completeness, culminating in a local product structure in the leaf space . They extend discrete results on uniformly compact center foliations to the flow setting, establish global -holonomy maps under suitable regularity, and derive quasi-shadowing properties in the quotient. The results have implications for transversely holomorphic partially hyperbolic flows on seven-dimensional manifolds and connect continuous-time dynamics with established discrete-time coherence theory. Overall, the work provides structural robustness—completeness, coherence, and local product structure—for partially hyperbolic flows under controlled holonomy and compactness assumptions.

Abstract

In this paper, we introduce the notion of dynamical coherence for a partially hyperbolic flow on a smooth compact manifold , and prove it under the assumption that there exists a compact foliation with trivial holonomy which integrates the subcenter distribution.
Paper Structure (18 sections, 42 theorems, 58 equations)

This paper contains 18 sections, 42 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Definitions
  3. Foliations
  4. Holonomy
  5. Compact foliations
  6. Partially hyperbolic flows
  7. Dynamical coherence
  8. Global $su\Phi$-holonomy maps
  9. Subcenter bunching
  10. Sketch of proof of the results
  11. General results on foliations transverse to flows
  12. Completeness of the center foliation
  13. Unstable projection
  14. Unstable projection distance and completeness of $E^c\oplus E^s$
  15. Completeness of the subcenter foliation
  16. ...and 3 more sections

Key Result

Theorem 2.6

Let $\mathcal{F}$ a compact $C^0$ foliation on a smooth manifold $M$. Then the following assertions are equivalent: If $M$ is compact, then the above properties are also equivalent to: Moreover if $\mathcal{F}$ has trivial holonomy, then the resulting leaf space $M/\mathcal{F}$ is a topological manifold.

Theorems & Definitions (90)

  • Definition 2.2
  • Definition 2.3
  • Definition 2.5
  • Theorem 2.6
  • Theorem 2.7: Generalized Reeb Stability
  • Corollary 2.7.1
  • Definition 2.8
  • Definition 2.9
  • Proposition 2.10
  • proof
  • ...and 80 more