Intersecting Legendrians and blow-ups

The purpose of this note is to describe the relationship between two classes of Legendre distributions. These two classes are distributions associated to an intersecting pair of Legendre submanifolds, introduced in [2] by analogy with intersecting Lagrangian distributions of Melrose and Uhlmann [8], and Legendre distributions associated to a fibred scattering structure introduced in [3]. We prove that, given appropriate symbolic orders, the first class is a proper subset of the second. We also give an example in two dimensions, which shows explicitly the relation between the two spaces in a simple setting.