The Simplest Branched Surfaces for a Foliation

Given a foliation F of a closed 3-manifold and a Smale flow φ transverse to F, we associate a “simplest” branched surface with the pair (F, φ), which is unique up to two combinatorial moves. We show that all branched surfaces constructed from F and φ can be obtained from the simplest model by applying a finite sequence of these moves chosen so that each intermediate branched surface also carries F. This is used to partition foliations transverse to the same flow into countably many equivalence classes.