Regularity and persistence in non-Weinstein Liouville geometry via hyperbolic dynamics

TL;DR

This paper investigates the geometric and dynamical properties of non-Weinstein Liouville structures derived from hyperbolic Anosov flows, establishing rigidity results and applications to partially hyperbolic dynamics.

Contribution

It introduces a framework connecting Anosov flows with non-Weinstein Liouville geometry, proving rigidity and regularity results in this novel setting.

Findings

Characterization of 4D non-Weinstein Liouville geometry via Mitsumatsu's examples

Establishment of $C^1$-persistence of the transverse skeleton

Applications to regularity of weak dominated bundles in hyperbolic flows

Abstract

We explore the construction of non-Weinstein Liouville geometric objects based on Anosov 3-flows, intoduced by Mitsumatsu, in the generalized framework of Liouville Interpolation Systems and non-singular partially hyperbolic flows. We study the subtle phenomena inherited from the regularity and persistence theory of hyperbolic dynamics in the resulting Liouville structures, and prove dynamical and geometric rigidity results in this context. Among other things, we show that Mitsumatsu's examples characterize 4-dimensional non-Weinstein Liouville geometry with 3-dimensional $C^1$-persistent transverse skeleton. We also draw applications to the regularity theory of the weak dominated bundles for non-singular partially hyperbolic 3-flows.