Katok's conjecture on unstable homogeneous flows

Let a homogeneous flow act on a compact homogeneous space, and let stability mean that the Lie derivative on smooth functions has closed range. Suppose the flow fails to be stable. A Liouvillean linear flow is a linear toral flow whose frequency vector fails the relevant Diophantine condition. The toral functions are the functions arising from the torus factor, and the orthogonal complement of their subspace consists of functions with zero average along each fiber of the projection.

Katok's conjecture. Every homogeneous flow on a compact homogeneous space that fails to be stable projects onto a Liouvillean linear flow on a torus. In this case, the flow is still stable on the orthogonal complement of the subspace of toral functions.

This is stated in the paper's open-problems section as a conjecture about the structure of non-stable homogeneous flows. The supplied status evidence records results for unipotent flows and nilflows, but does not resolve this broader compact homogeneous-flow conjecture.