Dynamical conditions for nowhere-scattered crossed products

Determine the dynamical conditions on an action of the free group F_2 on a compact Hausdorff space X that ensure the reduced crossed product C(X)\rtimes_r F_2 is nowhere scattered.

Background

The paper relates nowhere-scatteredness to pureness and shows that certain Bernoulli-shift crossed products associated with wreath products have elementary ideal-quotients, and therefore are not nowhere scattered.

In contrast, the paper does not determine which dynamical properties of an arbitrary compact Hausdorff F_2-action guarantee that its reduced crossed product is nowhere scattered. This remains an explicitly posed dynamical classification question.

References

In light of the results here and , we also ask the following.

Let $F_2\curvearrowright X$ be an action on a compact Hausdorff space. Under which dynamical conditions is $C(X)\rtimes_r F_2$ nowhere scattered?

— Nonamenable groups whose reduced group C*-algebras are not pure  (2609.10653 - Bell, 9 Sep 2026) in Section 2, immediately after the first concluding question