  The properties of uniform hyperbolicity and dominated splitting have been
introduced to study the stability of the dynamics of diffeomorphisms. One meets
difficulties when one tries to extend these definitions to vector fields and
Shantao Liao has shown that it is more relevant to consider the linear
Poincar\'e flow rather than the tangent flow in order to study the properties
of the derivative. In this paper we define the notion of singular domination,
an analog of the dominated splitting for the linear Poincar\'e flow which is
robust under perturbations. Based on this, we give a new definition of
multi-singular hyperbolicity which is equivalent to the one recently introduced
by Bonatti-da Luz. The novelty of our definition is that it does not involve
the blowup of the singular set and the renormalization cocycle of the linear
flows.
