Hyperbolic dynamics: uniform and beyond

Ever since the fundamental work of Peixoto and Smale on structural stability of discrete-time dynamics, the notion of hyperbolicity has been introduced into the field of dynamical systems. However, the global uniformly hyperbolic assumption (Anosov) is a rigid one, and researchers are looking for proper generalizations for this notion. In this talk, I will present some classic properties of uniform and non-uniform hyperbolicity, as well as some new generalizations introduced by Buzzi-Crovisier-Sarig in a recently published paper.