Measurable versions of the LS category on laminations

Abstract

We give two new versions of the LS category for the set-up of measurable laminations defined by Bermúdez. Both of these versions must be considered as "tangential categories". The first one, simply called (LS) category, is the direct analogue for measurable laminations of the tangential category of (topological) laminations introduced by Colman Vale and Macías Virgós. For the measurable lamination that underlies any lamination, our measurable tangential category is a lower bound of the tangential category. The second version, called the measured category, depends on the choice of a transverse invariant measure. We show that both of these "tangential categories" satisfy appropriate versions of some well known properties of the classical category: the homotopy invariance, a dimensional upper bound, a cohomological lower bound (cup length), and an upper bound given by the critical points of a smooth function.