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In broad terms, the goal of dynamics is to describe the long-term evolution of systems for which an ""infinitesimal"" evolution rule, such as a differential equation or the iteration of a map, is known.This book aims to put such recent developments in a unified perspective, and to point out open problems and likely directions for further progress. It is aimed willing to get a quick, yet broad, view of this part of dynamics. Main ideas, methods, and results are discussed, at variable degrees of depth, with references to the original works for details and complementary information.