Generalization of the Annulus Isotopy Theorem

Establish an analogue of the Annulus Isotopy Theorem for arbitrary decomposition surfaces satisfying suitable conditions, extending the result from product annuli to a broader class of decomposition surfaces.

Background

The paper proves an Annulus Isotopy Theorem showing that, under atoroidal and dynamical hypotheses, a product annulus can be isotoped into a position aligned with a pseudo-Anosov flow or a suitable further blowup. The author explicitly conjectures that comparable alignment results should hold for more general decomposition surfaces, which would extend the flow-version of the sutured decomposition framework beyond the annular case.