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ebsd2021:raissy [2021/09/10 13:35] – escolaebsd2021:raissy [2021/10/25 08:07] (current) – tahzibi
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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.\\ 
  
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 The **tentative plan** of the minicourse is as follows:\\ The **tentative plan** of the minicourse is as follows:\\
 1-  Introduction to parabolic Fatou components and the Leau-Fatou Flower Theorem in dimension 1\\ 1-  Introduction to parabolic Fatou components and the Leau-Fatou Flower Theorem in dimension 1\\
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 **References for the minicourse**: **References for the minicourse**:
  
-\bib{1} {\sc M. Astorg, X. Buff, R. Dujardin, H. Peters and J. Raissy:} {\sl A two-dimensional polynomial mapping with a wandering Fatou component} Ann. of Math. (2) 184 (2016), no. 1, 263--313.+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.\\
  
-\bib{2} {\sc E. Bedford, J. Smillie and T. Ueda:} {\sl Parabolic bifurcations in complex dimension 2.} Comm. Math. Phys. 350 (2017), no. 1, 1--29.+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\\
  
-\bib{3}{\sc X. Buff:} {\sl Wandering Fatou Component for Polynomials}  in {\bf KAWA Lectures Notes}, Annales de la faculté des sciences de Toulouse, S\’er. 6 (2018) 27/2, 445--475+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\\
  
-\bib{4} {\sc A. Douady:} {\sl 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.+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\\
  
-\bib{5} {\sc D. Sullivan} {\sl Quasiconformal Homeomorphisms and Dynamics I. Solution of the Fatou-Julia Problem on Wandering Domains.} Ann. of Math. (2) {\bf 122} (1985), no. 3, 401--418.+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:**\\ **Prerequisites:**\\
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 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.\\ 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**\\ +**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:raissi2|Tema 2]]+[[ebsd2021:raissi2|Mapas racionais na esfera de Riemann]]
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-[[ebsd2021:raissi3|Tema 3]]+[[ebsd2021:raissi3|Teoria local: dinâmica perto de pontos periódicos]]
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-[[ebsd2021:raissi4|Tema 4]]+[[ebsd2021:raissi4|Teoria global: conjuntos de Julia e de Fatou e suas propriedades]] 
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 +[[ebsd2021:raissy5|Wandering domains for polynomials in higher dimension]]
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ebsd2021/raissy.1631291714.txt.gz · Last modified: 2021/09/10 13:35 by escola