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

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.