Finite jet determination of local analytic CR automorphisms and their parametrization by 2-jets in the finite type case

We show that germs of local real-analytic CR automorphisms of areal-analytic hypersurface M in $mathbb{C}$² at a point p M are uniquelydetermined by their jets of some finite order at p if and only if M isnot Levi-flat near p. This seems to be the first necessary and sufficientresult on finite jet determination and the first result of this kind inthe infinite type case.If M is of finite type at p, we prove a stronger assertion: the local real-analytic CR automorphisms of M fixing p are analytically parametrized (and hence uniquely determined) by their 2-jets at p.This result is optimal since the automorphisms of the unit sphere arenot determined by their 1-jets at a point of the sphere. The finitetype condition is necessary since otherwise the needed jet order canbe arbitrarily high [Kow1,2], [Z2]. Moreover, we show, by an example,that determination by 2-jets fails for finite type hypersurfaces alreadyin $mathbb{C)$³.We also give an application to the dynamics of germs of localbiholomorphisms of $mathbb{C)$².