On C1-robust transitivity of volume-preserving flows

TL;DR

This paper proves that divergence-free, C1-robustly transitive vector fields lack singularities, and for C4 fields, their linear Poincare flow admits a dominated splitting, advancing understanding of volume-preserving dynamical systems.

Contribution

It establishes the absence of singularities in C1-robust transitive volume-preserving flows and demonstrates dominated splitting for C4 vector fields, providing new insights into their structure.

Findings

No singularities in divergence-free, C1-robustly transitive vector fields.

Existence of dominated splitting for C4 vector fields' linear Poincare flow.

Enhanced understanding of the structure of volume-preserving flows.

Abstract

We prove that a divergence-free and C1-robustly transitive vector field has no singularities. Moreover, if the vector field is C4 then the linear Poincare flow associated to it admits a dominated splitting over M.