Horizontality of partially hyperbolic foliations

TL;DR

The paper classifies Seifert manifolds that admit partially hyperbolic diffeomorphisms, showing a circle bundle over a higher-genus surface supports such a diffeomorphism iff it is nilmanifold, double covered by a nilmanifold, or finitely covered by the unit tangent bundle of a hyperbolic orbifold; it proves cs/cu foliations are horizontal and that, for the unit tangent bundle of a higher-genus surface, every partially hyperbolic diffeomorphism is a collapsed Anosov flow. The authors develop and refine branching foliations, construct approximating foliations in ideal positions, and use center-branching analyses together with averaged vector fields and Poincaré–Hopf arguments to rule out non-horizontal configurations. A central methodological advance is the reduction to a horizontal regime via Brittenham-type results, followed by a global averaging construction whose zeros and indices conflict with the Euler characteristic of the base surface, yielding a contradiction unless horizontality holds. The results unify and extend prior works on partially hyperbolic dynamics on 3-manifolds, linking circle bundles, unit tangent bundles, and collapsed Anosov flow models, and provide a complete geometric-dynamical picture for these manifolds.

Abstract

We show exactly which Seifert manifolds support partially hyperbolic dynamical systems. In particular, a circle bundle over a higher-genus surface supports a partially hyperbolic system if and only if it supports an Anosov flow. We also show for these systems that the center-stable and center-unstable foliations can be isotoped so that their leaves are transverse to the circle fibering. As a consequence, every partially hyperbolic system defined on the unit tangent bundle of a higher-genus surface is a collapsed Anosov flow.

Figures (6)

• Figure 1: A graphical depiction of \ref{['lemma:hypdisk']}. The horizontal line here is the geodesic $\ell$ in the Poincaré disk model of $\mathbb{H}^2$ and the two dashed curves are at a fixed distance from $\ell.$ The arrows depict the projection $\pi_\ell$ along geodesics perpendicular to $\ell.$ The shaded region is the topological disk $D_0$ which has boundary $\alpha_+ \cup \alpha_- = \alpha.$
• Figure 2: The map $\hat{f} : \hat{M} \to T \mathbb{H}^2$ at a point $p \in \hat{M}.$ This map is quotiented to produce the induced map $f : M \to T S.$
• Figure 3: A center segment $J_n$ inside of a vertical cs leaf $L_n$ as in the proof of \ref{['lemma:goodtime']}. The top of the figure is identified with the bottom, so that each vertical line represents a circle in the circle fibering.
• Figure 4: A surface with boundary for which \ref{['thm:poincarehopf']} applies. The piecewise $C^1$ boundary has eight pieces, four of which are transverse to the flow. The boundary has two half-saddles and we assume that there are no other fixed points. The Euler characteristic is therefore $\chi(S) = \tfrac{1}{2} 4 + (\tfrac{-1}{2}) + (\tfrac{-1}{2}) = 1$.
• Figure 5: A depiction of the curve $J_z$ and the corresponding surface $S_z$. The vertical line through the point $p$ depicts the circle $\{0\} \times \{0\} \times S^1$ where the top and bottom points are identified. Similarly, the top and bottom points of $C$ are identified.

Theorems & Definitions (96)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Theorem 1.6: Brittenham
• Theorem 1.7
• Theorem 1.8
• Proposition 1.9
• Theorem 2.1: Burago--Ivanov