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In the first part of this text we give a survey of the properties satisfiedby the C1-generic conservative diffeomorphisms of compact surfaces. The mainresult that we will discuss is that a C1-generic conservative diffeomorphism ofa connected compact surface is transitive. It is obtain as a consequence of aconnecting lemma for pseudo-orbits. In the last parts we expose some recentdevelopments of the C1-perturbation technics and the proof of this connectinglemma. We are not aimed to deal with technicalities nor to give the finestavailable versions of these results. Besides this theory exists also in higherdimension and in the non-conservative setting, we restricted the scope of thispresentation to the conservative case on surfaces, since it offers somesimplifications which allow to explain in an easier way the main ideas of thesubject.

Perturbations of C1-diffeomorphisms and dynamics of generic conservative diffeomorphisms of surface