Elliptic Fixed Points with an Invariant Foliation: Some Facts and More Questions

@article{Chenciner2022EllipticFP,
  title={Elliptic Fixed Points with an Invariant Foliation: Some Facts and More Questions},
  author={Alain Chenciner and David Sauzin and Shanzhong Sun and Qiaoling Wei},
  journal={Regular and Chaotic Dynamics},
  year={2022},
  volume={27},
  pages={43-64}
}
We address the following question: let $$F:(\mathbb{R}^{2},0)\to(\mathbb{R}^{2},0)$$ be an analytic local diffeomorphism defined in the neighborhood of the nonresonant elliptic fixed point 0 and let $$\Phi$$ be a formal conjugacy to a normal form $$N$$ . Supposing $$F$$ leaves invariant the foliation by circles centered at $$0$$ , what is the analytic nature of $$\Phi$$ and $$N$$ ? 

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