Negative-divergence obstruction to invariant tori ($\nabla \cdot F<0$)

A smooth embedded invariant two-torus in three-dimensional Euclidean phase space bounds a solid torus of positive volume. The vector field is tangent to its boundary, so its normal flux is zero. If the divergence is strictly negative throughout the enclosed volume, the divergence theorem gives a strictly negative flux, a contradiction. Thus a globally negative-divergence three-dimensional vector field has no such quasiperiodic invariant torus. This argument is stronger than merely noting contraction of transported phase volume; an attractor's own zero volume is not a contradiction.