The Anosov-Smale theory of uniform hyperbolicity provides guidance for the dynamics of many systems beyond those satisfying exactly its axioms. Hence it is a recurring theme in dynamics to find «non-uniformly hyperbolic settings» that are as general as possible, imply the classical consequences, and can be easily checked. I will report on such a new notion, called «strong positive recurrence (SPR)» and developed with Sylvain CROVISIER and Omri SARIG.