Geometric normalization

Alain Chenciner; David Sauzin; Qiaoling Wei

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arXiv · 2506.11580

Geometric normalization

Alain Chenciner·David Sauzin·Qiaoling Wei

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Abstract

For a local analytic diffeomorphism of the plane with an irrational elliptic fixed point at 0, we introduce the notion of ``geometric normalization'', which includes the classical formal normalizations as a special case: it is a formal conjugacy to a formal diffeomorphism which preserves the foliation by circles centered at 0. We show that geometric normalizations, despite of non-uniqueness, correspond in a natural way to a unique formal invariant foliation. We show, in various contexts, generic results of divergence for the geometric normalizations, which amount to the generic non-existence of any analytic invariant foliation.

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Alain Chenciner, David Sauzin, Qiaoling Wei. 2025-06-13. Geometric normalization. https://arxiv.org/abs/2506.11580

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