Weak hyperbolic structures and robust properties of diffeomorphisms and flows

Abstract

A property of a dynamical system is called $C^r$-robust if it holds on a $C^r$-open set of systems. For diffeomorphisms or for non-singular flows, there are many results relating $C^1$-robust properties and global structures of the dynamics, as hyperbolicity, partial hyperbolicity, dominated splitting. However, a difficulty appears when a robust property of a flow holds on a set containing recurrent orbits accumulating a singular point. This phenomenon is now mainly understood in dimension 3, but is still wide open in higher dimensions. Here, we propose a general procedure for adapting the usual hyperbolic structures to the singularities, opening the door for bypassing the difficulty of the coexistence of singular and regular orbits. In particular, this new definition allows us to adapt the proof in [15] to get a characterization of star flows on a $C^1$-open and dense set.