Contributions to the Geometric and Ergodic Theory of Conservative Flows

TL;DR

This paper establishes a dichotomy for volume-preserving flows, showing that almost all points have zero Lyapunov exponents or the flow admits a dominated splitting, with implications for ergodicity and hyperbolicity.

Contribution

It proves a new dichotomy for C1-residual volume-preserving flows and links the absence of elliptic periodic orbits to dominated splittings and hyperbolic approximations.

Findings

Almost every point has zero Lyapunov exponents or the flow admits a dominated splitting.

Flows not approximable by elliptic periodic orbits have dominated splittings on full measure sets.

Volume-preserving, stably ergodic flows can be C1-approximated by non-uniformly hyperbolic flows.

Abstract

We prove the following dichotomy for vector fields in a C1-residual subset of volume-preserving flows: for Lebesgue almost every point all Lyapunov exponents equal to zero or its orbit has a dominated splitting. As a consequence if we have a vector field in this residual that cannot be C1-approximated by a vector field having elliptic periodic orbits, then, there exists a full measure set such that every orbit of this set admits a dominated splitting for the linear Poincare flow. Moreover, we prove that a volume-preserving and C1-stably ergodic flow can be C1-approximated by another volume-preserving flow which is non-uniformly hyperbolic.