The dynamics of quasi-isometric foliations

TL;DR

This paper investigates the properties of quasi-isometric foliations in partially hyperbolic systems, establishing conditions for global product structure, holonomy absence, and homeomorphism of center leaves.

Contribution

It proves that quasi-isometric stable, center, and unstable foliations imply global product structure and center leaf homeomorphism in certain dynamical systems.

Findings

Quasi-isometric foliations imply global product structure.

Center foliation is without holonomy under certain conditions.

All center leaves are homeomorphic when global product structure exists.

Abstract

If the stable, center, and unstable foliations of a partially hyperbolic system are quasi-isometric, the system has Global Product Structure. This result also applies to Anosov systems and to other invariant splittings. If a partially hyperbolic system on a manifold with abelian fundamental group has quasi-isometric stable and unstable foliations, the center foliation is without holonomy. If, further, the system has Global Product Structure, then all center leaves are homeomorphic.