Divergence of geometric normalization for an elliptic fixed point in the plane

SpeakerQiaoling Wei
Date: 12.03.24 at 14:00

Abstract: Classically, for a local analytic diffeomorphism F of (R^2,0) with a non-resonant elliptic fixed point (eigenvalues exp(\pm2\pi i\omega) with \omega real irrational), one can find formal normalizations, i.e. formal conjugacies to a formal diffeomorphism invariant under the group of rotations. Less demanding is the notion of a “geometric normalization” that we introduce: this is a formal conjugacy to a formal diffeomorphism which maps any circle centered at 0 to a circle centered at 0. Geometric normalizations are not unique, but they correspond in a natural way to a unique formal invariant foliation. We then show that, generically, all geometric normalizations are divergent, so there is no analytic invariant foliation. The talk is based on joint works with Alain Chenciner, David Sauzin.