The presence of emergent fractal clusters in spin models’ percolative scaffoldings, as well as in pure percolation models, is nowadays well recognized. Also appreciated is their role in the critical behavior. Recalling a 1978 counterexample to Nelson’s 1962 conjecture, I shall present another example of fractality (fractal singularities) with striking implications for measure-preserving flows in d≥3 dimensions.

The counterexamples to the proposed condition for the existence and uniqueness of Lagrangian flows are considerably simpler than what it took to address analogous questions for the Euler equations. Yet there are certain similarities in the constructions of vector fields with singular behavior which nevertheless meet the tested admissibility criteria.