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

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