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

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Abstract

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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Notes

  1. Thanks to Abed Bounemoura for insisting on this.

  2. Consider \(\Psi\) as a function of two independent variables \(z\) and \(\bar{z}\).

  3. Thanks to Ricardo Pérez – Marco for this reference.

  4. A translated orbit is an orbit whose image under \(F\) is obtained by a radial translation by some constant. They exist independently of the hypothesis that \(F\) preserves the foliation \({\cal F}_{0}\).

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ACKNOWLEDGMENTS

The authors thank Jacques Féjoz, Abed Bounemoura and Ricardo Pérez-Marco for fruitful questions and discussions.

Funding

The first two authors thank Capital Normal University for its hospitality.

The third author is partially supported by the National Key R&D Program of China (2020YFA0713300), NSFC (Nos. 11771303, 12171327, 11911530092, 11871045).

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Authors and Affiliations

  1. IMCCE, Observatoire de Paris, PSL Research University, CNRS, av. Denfert-Rochereau 77, 75014, Paris, France

    Alain Chenciner & David Sauzin

  2. University Paris 7, F75014, Paris, France

    Alain Chenciner

  3. Department of Mathematics, Capital Normal University , 100048, Beijing, China

    Shanzhong Sun & Qiaoling Wei

  4. Academy for Multidisciplinary Studies, Capital Normal University, 100048, Beijing, China

    Shanzhong Sun

Authors
  1. Alain Chenciner
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  2. David Sauzin
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  3. Shanzhong Sun
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  4. Qiaoling Wei
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Corresponding authors

Correspondence to Alain Chenciner, David Sauzin, Shanzhong Sun or Qiaoling Wei.

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The authors declare that they have no conflicts of interest.

Additional information

Dedicated to the memory of our friend and colleague Alexey Borisov

MSC2010

37E30, 37G05

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Chenciner, A., Sauzin, D., Sun, S. et al. Elliptic Fixed Points with an Invariant Foliation: Some Facts and More Questions. Regul. Chaot. Dyn. 27, 43–64 (2022). https://doi.org/10.1134/S1560354722010063

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  • DOI: https://doi.org/10.1134/S1560354722010063

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