A Pasting Lemma I: the case of vector fields

TL;DR

This paper introduces a new perturbation lemma for conservative vector fields, demonstrating density of smooth volume-preserving fields, and establishes structural properties of robustly transitive conservative flows, with implications for ergodicity and dynamical stability.

Contribution

It presents a novel pasting lemma for conservative systems, proves density of smooth volume-preserving vector fields, and characterizes robust transitivity and ergodic properties in conservative dynamics.

Findings

C^∞ volume preserving vector fields are C^1-dense in C^1 volume preserving vector fields.

C^1 robustly transitive conservative flows in 3D are Anosov.

No geometrical Lorenz-like sets exist for conservative flows.

Abstract

We prove a perturbation (pasting) lemma for conservative (and symplectic) systems. This allows us to prove that $C^{\infty}$ volume preserving vector fields are $C^1$-dense in $C^{1}$ volume preserving vector fields (After the conclusion of this work, Ali Tahzibi pointed out to us that this result was proved in 1979 by Carlos Zuppa, although his proof is different from ours.). Moreover, we obtain that $C^1$ robustly transitive conservative flows in three-dimensional manifolds are Anosov and we conclude that there are no geometrical Lorenz-like sets for conservative flows. Also, by-product of the version of our pasting lemma for conservative diffeomorphisms, we show that $C^1$-robustly transitive conservative $C^2$-diffeomorphisms admits a dominated splitting, thus solving a question posed by Bonatti-Diaz-Pujals. In particular, stably ergodic diffeomorphisms admits a dominated splitting.