On the notions of singular domination and (multi-)singular hyperbolicity

The properties of uniform hyperbolicity and dominated splitting have been introduced to study the stability of the dynamics of diffeomorphisms. One meets difficulties when trying to extend these definitions to vector fields and Shantao Liao has shown that it is more relevant to consider the linear Poincaré 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 Poincaré 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 and da Luz (2017). The novelty of our definition is that it does not involve the blow-up of the singular set and the rescaling cocycle of the linear flows.