= Solution
The Lorenz <vector field> has constant <divergence> $\nabla\cdot F=-\sigma-1-b$. <Liouville formula for a fundamental matrix> for the <derivative> of the flow therefore gives
$$
\boxed{\det D\Phi_t=e^{-(\sigma+1+b)t},\qquad V(t)=V(0)e^{-(\sigma+1+b)t}.}
$$
This concerns transported three-dimensional phase volume, not the Euclidean distance between each pair of trajectories.
A quasiperiodic orbit with the usual smooth two-frequency <torus> closure would lie on an invariant embedded two-torus. That <torus> bounds a solid <torus> of positive finite volume in $\mathbb R^3$. Since the <vector field> is tangent to its boundary, its outward flux is zero, contradicting the <divergence theorem>, whose volume integral is $-(\sigma+1+b)V<0$. This <negative-divergence obstruction to invariant tori> excludes such quasiperiodic tori. The argument uses global negative <divergence> and the enclosed volume; saying only that an attracting <torus> has zero volume would not exclude it.
At an <equilibrium> the sum of <eigenvalues> is $-(\sigma+1+b)<0$, so they cannot all have positive real parts: no <equilibrium> is a fully repelling source. For a <periodic orbit> of period $T$, one <Floquet multiplier> equals one in the flow direction. The product of the other two is $e^{-(\sigma+1+b)T}<1$, so they cannot both have modulus greater than one: no <periodic orbit> is fully repelling either. Saddle <equilibria> and saddle <periodic orbits> remain possible; contraction of volume does not imply contraction in every direction.
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