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This text deals with birationnal diffeomorphisms of real algebraic surfaceswhich have simple real dynamics and rich complex dynamics. We give an exampleof such a transformation on P^1xP^1, then we show that this situation isexceptional and implies strong conditions on both the topology of the reallocus and the real dynamics.

Sur la dynamique des diff\'eomorphismes birationnels des surfaces alg\'ebriques r\'eelles : ensemble de Fatou et lieu r\'eel