Detecting when a nonsingular flow is transverse to a foliation
In [\textit{S. Goodman} and \textit{S. Shields}, Topology Appl. 154, No.~16, 2962--2975 (2007; Zbl 1143.57014)] it was shown that when a flow \(\phi\) has no self-return disk (that is, a disk transverse to \(\phi\) that flows continuously into its own interior), then a simple algorithm for successively modifying any branched surface transverse to \(\phi\) will eventually produce a branched surface carrying a foliation \(F\) precisely when \(F\) is transverse to \(\phi.\) In the present paper it is shown that this algorithm also works when \(\phi\) has self-return disks. Concequently, a procedure is found that can be used to determine whether or not any branched surface produced by the algorithm carries a foliation. In particular, a process is described that allows to modify any branched surface in order to produce an essential branched surface that carries a foliation if and only if the original does. Algorithms in [\textit{I. Agol} and \textit{T. Li}, Geom. Topol. 7, 287--309 (2003; Zbl 1037.57008)] can then be applied to determine whether or not this new branched surface carries a foliation. Hence the author obtains algorithmic means for detecting flows with transverse foliations. Furthermore, the author proves that for any \(\delta>0,\) one can find a positive integer \(K\) such that if the branched surface produced at the \(K\)-th stage of our algorithm does not carry a foliation transverse to \(\phi,\) then there are no foliations that remain a bounded distance of at least \(\delta\) from \(\phi.\) If, on the other hand, this branched surface does carry a foliation, then the algorithm suggested by the author will detect that it does.
scientific article; zbMATH DE number 5858919