Generalization of multi-specializations and multi-asymptotics

The aim of this paper is to give a new description of the geometry appearing in the multi-specialization along a general family of submanifolds of a real analytic manifold (including some important cases as clean intersection or a simultaneously linearizable family of Lagrangian submanifolds in a cotangent bundle) and then, to extend several properties of the multi-specialization. The notion of multi-asymptotic expansions is also extended. In the local model more general cases are studied: locally we can construct new sheaves of multi-asymptotically developable functions closely related with asymptotics along a subvariety with a simple singularity such as a cusp.