On transversely holomorphic partially hyperbolic flows

TL;DR

This paper investigates transversely holomorphic partially hyperbolic flows, showing that under certain conditions, they project to well-understood transversely holomorphic Anosov flows, linking complex dynamics with hyperbolic geometry.

Contribution

It establishes a classification result for seven-dimensional flows with specific integrability and holonomy conditions, connecting them to known hyperbolic and toral automorphism flows.

Findings

Flow projects to transversely holomorphic Anosov flow

Topologically transitive flows are orbit equivalent to classical hyperbolic models

Results connect complex holomorphic dynamics with hyperbolic geometry

Abstract

In this paper, we study transversely holomorphic partially hyperbolic flows, i.e. those whose holonomy pseudo-group is given by biholomorphic maps. We prove in the seven-dimensional case that under the assumption that the subcenter distribution is integrable to a flow invariant compact foliation with trivial holonomy, then the flow projects, by a smooth fiber bundle map, to a transversely holomorphic Anosov flow on a smooth five-dimensional manifold which is, in case of topological transitivity, either $C^\infty$ orbit equivalent to the suspension of a hyperbolic automorphism of a complex torus, or, up to finite covers, $C^\infty$-orbit equivalent to the geodesic flow of a compact hyperbolic manifold.