# 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}
}
• Published 15 November 2021
• Mathematics
• Regular and Chaotic Dynamics
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 $$N$$ ?

## References

SHOWING 1-10 OF 15 REFERENCES

### On the divergence of Birkhoff Normal Forms

It is well known that a real analytic symplectic diffeomorphism of the 2 d $2d$ -dimensional disk ( d ≥ 1 $d\geq 1$ ) admitting the origin as a non-resonant elliptic fixed point can be formally

### Siegel discs, Herman rings and the Arnold family

We show that the rotation number of an analytically linearizable element of the Arnold family fa,b(X) = x + a + bsin(27rx) (mod 1), a, b c IR, O < b < 1/(27r), satisfies the Brjuno condition.

### Perturbing a Planar Rotation: Normal Hyperbolicity and Angular Twist

In generic two-parameter families of local diffeomorphisms of the plane unfolding a local diffeomorphism with an elliptic fixed point, the tension between radial (hyperbolic) and tangential

### Convergence or generic divergence of the Birkhoff normal form

We prove that the Birkhoff normal form of hamiltorlian flows at a nonresonant singular point with given quadratic part is always convergent or generically divergent. The same result is proved for the

### Solution complète au problème de Siegel de linéarisation d'une application holomorphe au voisinage d'un point fixe (d'après J.-C. Yoccoz)

© Société mathématique de France, 1992, tous droits réservés. L’accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/ Publications/Asterisque/) implique l’accord avec les

### Mapping the circle onto itself

• (Russian), Izv. Akad. Nauk SSSR Ser. Mat
• 1961

• Russian)