Abstract
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 (elliptic) behaviour gives rise to phenomena where the whole wealth of the area preserving case is unfolded along some direction of the parameter space.
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Notes
- 1.
Beware that the notation F(ζ) does not mean that F is complex analytic, its expression depends on ζ and \(\overline \zeta \).
- 2.
We shall not give a formal definition of this word; it means essentially that what is described is the general situation and that only special hypotheses could prevent the description from being correct.
- 3.
Roughly speaking this means that any attraction or repulsion normal to the curve under the iterates of F μ dominates any attraction or repulsion inside the curve; this condition insures the robustness of the curve.
- 4.
See section 1.4 of [7] for a brief introduction.
- 5.
In order to avoid too cumbersome notations we still call z the transformed coordinate H 3(z).
- 6.
All figures in this section are reproduced with the kind permission of Publications mathématiques de l’IHÉS.
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Acknowledgements
This text was written at the occasion of lectures given at IRMA (Université de Strasbourg), Southwest Jiaotong University (Chengdu), Tsinghua University and Capital Normal University (Beijing). The author warmly thanks the colleagues in these institutions who gave him the occasion to revisit these topics. Finally, thanks to both anonymous referees for a thorough reading which led to some clarifications and the elimination of typos.
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Chenciner, A. (2019). Perturbing a Planar Rotation: Normal Hyperbolicity and Angular Twist. In: Dani, S.G., Papadopoulos, A. (eds) Geometry in History. Springer, Cham. https://doi.org/10.1007/978-3-030-13609-3_11
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