  Using the definition of entropy of a family of increasing distances on a
compact metric set given in [10] we introduce a notion of Finsler entropy for
smooth distributions and Stefan-Sussmann foliations. This concept generalizes
most of classical topological entropy on a compact Riemannian manifold : the
entropy of a flow ([9]), of a regular foliation ([11]), of a regular
distribution ([5]) and of a geometrical structure ([22]). The essential results
of this paper is the nullity of the Finsler entropy for a controllable
distribution and for a singular Riemannian foliation.
