On the Integrability of codimension-one invariant subbundles of partially hyperbolic skew-products

TL;DR

This paper investigates the integrability of invariant subbundles in partially hyperbolic skew-products on tori, showing conditions under which the hyperbolic structure is integrable and not contact, with specialized results in three dimensions.

Contribution

It establishes a class of maps ensuring integrable hyperbolic structures in partially hyperbolic skew-products, extending understanding of their geometric properties.

Findings

Existence of maps b3 with integrable hyperbolic structures

In dimension 3, characteristic foliations provide a simpler proof

The results imply these systems are not contact diffeomorphisms

Abstract

We prove there is a class of maps $\gamma:\mathbb{T}^{2n}\rightarrow\mathbb{S}^1$ such that a conservative dynamically coherent partially hyperbolic skew-product on $\mathbb{T}^{2n}\times\mathbb{S}^1$ with fixed hyperbolic dynamics on the base and rotation by angle $\gamma$ acting on the fibers have integrable hyperbolic structure which also implies in particular that they are not contact diffeomorphisms. In dimension $3$, we prove the same result using a standard technique in Contact Geometry, namely, that of \emph{characteristic foliations}, which gives a simple proof of the result but with more tight restrcitions to the map $\gamma$.