✎ ebsd2021:raissy []

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====== Wandering domains for polynomials in higher dimension ======

**Short abstract**: We will present the proof of the existence of a wandering domain for a polynomial endomorphism $\mathbb{C}^2$. The idea is to present the proof in such a way to emphasize the key steps in the construction and potential applications.

**Extended abstract**: The filled-in Julia set $K_f$ of a polynomial map $f\colon\mathbb{C} \to \mathbb{C}$ is the set of points with bounded orbit under iteration of $f$. The Non Wandering Theorem proved by Sullivan in the 1980’s asserts that every connected component of the interior of $K_f$ is eventually periodic.
The goal of the mini-course is to show that this result does not hold for polynomials maps $F\colon\mathbb{C}^2\to\mathbb{C}^2$. More precisely, we will show that if
$$
F(z,w)= \left(z - z^2, w + w^2 + aw^3 + {\pi^2\over 4} z \right)
$$
with $a < 1$ sufficiently close to $1$, then $F$ admits a wandering Fatou component.
The proof uses techniques of parabolic implosion for skew-products. We will emphasize the key steps in the construction and give further developments and applications.\\

The **tentative plan** of the minicourse is as follows:\\
1-  Introduction to parabolic Fatou components and the Leau-Fatou Flower Theorem in dimension 1\\
2-  Introduction to parabolic implosion in complex dimension 1\\
3-  Statement of the main result and begin of proof\\
4   End of proof of main result and further developments\\

**References for the minicourse**:

1- M. Astorg, X. Buff, R. Dujardin, H. Peters and J. Raissy: //A two-dimensional polynomial mapping with a wandering Fatou component,// Ann. of Math. (2) 184 (2016), no. 1, 263--313, https://arxiv.org/pdf/1411.1188.pdf.\\

2- E. Bedford, J. Smillie and T. Ueda: //Parabolic bifurcations in complex dimension 2//, Comm. Math. Phys. 350 (2017), no. 1, 1--29, https://arxiv.org/pdf/1208.2577.pdf\\

3- X. Buff: //Wandering Fatou Component for Polynomials,//  in KAWA Lectures Notes, Annales de la faculté des sciences de Toulouse, Ser. 6 (2018) 27/2, 445--475, http://www.numdam.org/item/10.5802/afst.1575.pdf\\

4- A. Douady: //Does a Julia set depend continuously on the polynomial?// Complex dynamical systems (Cincinnati, OH, 1994), 91--138, Proc. Sympos. Appl. Math., 49, Amer. Math. Soc., Providence, RI, 1994\\

5- D. Sullivan //Quasiconformal Homeomorphisms and Dynamics I. Solution of the Fatou-Julia Problem on Wandering Domains//, Ann. of Math. (2) 122 (1985), no. 3, 401--418, http://www.math.stonybrook.edu/~bishop/classes/math627.S13/Sullivan-1985-Nonwandering.pdf\\

**Prerequisites:**\\
1- Basic complex analysis in dimension 1\\
2- The Riemann sphere: its complex structures and the description of its group of holomorphic automorphisms\\
3- Basic discrete holomorphic dynamics on the Riemann sphere: local theory\\
4- Basic discrete holomorphic dynamics on the Riemann sphere: Fatou/Julia sets and their basic properties, statement of Fatou’s Classification of invariant Fatou Components, statement of Sullivan’s Non Wandering Theorem.\\

**Reference for the prerequisites (English)**\\
1-Carne, Geometry and Groups https://www.dpmms.cam.ac.uk/~tkc/GeometryandGroups/GeometryandGroups.pdf\\
2-Carleson and Gamelin, Complex Dynamics, https://zr9558.files.wordpress.com/2013/11/complex-dynamics-carleson.pdf\\
3-Milnor, Dynamics in one complex variable, https://arxiv.org/pdf/math/9201272.pdf

**Referencias para pre-requisitos (Português)**\\
Lomonaco, Notas de aulas, https://sites.google.com/view/lunalomonaco/teaching?authuser=0

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[[ebsd2021:raissi1|Preliminares em Análise Complexa]]
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[[ebsd2021:raissi2|Mapas racionais na esfera de Riemann]]
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[[ebsd2021:raissi3|Teoria local: dinâmica perto de pontos periódicos]]
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[[ebsd2021:raissi4|Teoria global: conjuntos de Julia e de Fatou e suas propriedades]]
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