Integrability of C^1 invariant splittings

TL;DR

This paper establishes new conditions under which C^1 invariant splittings are integrable, including volume domination and dynamical domination, with specific results for 2-dimensional decompositions on 3-manifolds.

Contribution

It introduces novel criteria for integrability of C^1 invariant splittings, extending known results to broader settings and specific cases like volume-preserving diffeomorphisms.

Findings

2D invariant decompositions on 3-manifolds are uniquely integrable under volume domination.

Volume-preserving diffeomorphisms require only dynamical domination for integrability.

New conditions generalize previous integrability criteria for invariant splittings.

Abstract

We derive some new conditions for integrability of dynamically defined C^1 invariant splittings in arbitrary dimension and co-dimension. In particular we prove that every 2-dimensional C^1 invariant decomposition on a 3-dimensional manifold satisfying a volume domination condition is uniquely integrable. In the special case of volume preserving diffeomorphisms we show that standard dynamical domination is already sufficient to guarantee unique integrability.