From \(2N\) to Infinitely Many Escape Orbits

In this short note, we prove that singular Reeb vector fields associated with generic \(b\)-contact forms on three dimensional manifolds with compact embedded critical surfaces have either (at least) \(2N\) or an infinite number of escape orbits, where \(N\) denotes the number of connected components of the critical set. In case where the first Betti number of a connected component of the critical surface is positive, there exist infinitely many escape orbits. A similar result holds in the case of \(b\)-Beltrami vector fields that are not \(b\)-Reeb. The proof is based on a more detailed analysis of the main result in [19].