On vector fields as generators of flows: A counterexample to Nelson’s conjecture


A construction is given for a collection of examples of divergenceless vector fields with integral curves along which, in a finite time, fibers are contracted to points. The examples, which include a uniformly bounded vector field, are used to prove that, contrary to a conjecture due to E. Nelson, no $L^p$ condition on the vector fields is sufficient, in $\mathbf{R}^d$, $d\gg 3$, for the existence or uniqueness of a generated measure preserving point flow.


Michael Aizenman