Krystyna Kuperberg: Preserving measure in differentiable dynamical systems

There are clear obstacles in constructing nonsingular smooth dynamical systems that preserve measure.

In 1996, G. Kuperberg gave an example of a measure preserving $C^1$ flow on $S^3$ with no compact orbits. The construction is based on Schweitzer's counterexample to the Seifert conjecture asserting that a flow on $S^3$ must posses a compact orbit. The minimal sets in his construction come in pairs and are of topological dimension one. In a similar fashion, V. Ginzburg constructed in 2007 a $C^1$ Hamiltonian flow on $S^3$ (the Hamiltonian on $\mathbb{R}^4$ is $C^2$) with no compact orbits.

The method that produces $C^r$, $r\geq 0$, and $C^\infty$ counterexamples to the Seifert conjecture on $S^3$ does not lead to a measure preserving flow if the differentiability is at least $C^1$, but there seem to be no obstacles to modify the example to a measure preserving $C^0$ flow with only one minimal set.

G. Kuperberg proved that every boundaryless 3-manifold possesses a nonsingular, measure preserving flow with a discrete set of periodic orbits. His method can be improved to have an additional condition that the closure of every orbit is compact, i.e., the orbits are bounded. It is not known whether one can construct such examples with orbits uniformly bounded.