The minimal foliation criterion for stable Bernoulli dynamics

Let $M$ be a compact manifold, let $m$ denote Lebesgue measure, and let $\operatorname{Diff}^{1}_{m}(M)$ be the space of $C^1$ Lebesgue-measure-preserving diffeomorphisms. An invariant expanding or contracting foliation is minimal if every leaf is dense, and stable Bernoulli means that the Bernoulli property persists throughout a suitable neighborhood in the volume-preserving $C^1$ topology. Minimal foliation criterion. Generically in $\operatorname{Diff}^{1}_{m}(M)$, the existence of a minimal invariant expanding or contracting foliation implies stable ergodicity, and even stable Bernoulli dynamics. The source proposes this as a more modest generic mechanism and does not report a resolution.