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September 29, 2025Open Access

Horizontality of partially hyperbolic foliations

Key Points

Partially hyperbolic systems exist in Seifert manifolds that support Anosov flows, enhancing our understanding of dynamical systems.
A circle bundle over a higher-genus surface supports a partially hyperbolic system if and only if it includes an Anosov flow.
The center-stable and center-unstable foliations in these systems can be adjusted to align with circle fiberings, revealing structural properties.
Every partially hyperbolic system on the unit tangent bundle of a higher-genus surface behaves like a collapsed Anosov flow, showcasing significant dynamical features.

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.

Horizontality of partially hyperbolic foliations

Key Points

Abstract

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