Differences

This shows you the differences between two versions of the page.

--- ebsd2021:raissy [2021/09/10 13:22] – escola
+++ ebsd2021:raissy [2021/10/25 08:07] (current) – tahzibi
@@ Line 1: / Line 1: @@
-Title: 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.
+<WRAP center round box 100%>
+[[ebsd2021:participantsraissy|Comments and Questions of participants]]
+</WRAP>
+The **tentative plan** of the minicourse is as follows:\\
+-  Introduction to parabolic Fatou components and the Leau-Fatou Flower Theorem in dimension 1\\
+-  Introduction to parabolic implosion in complex dimension 1\\
+-  Statement of the main result and begin of proof\\
+   End of proof of main result and further developments\\
-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.
+**References for the minicourse**:
-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:
+- 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.\\
-- Introduction and basic properties of holomorphic dynamics and Fatou components.
+- 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\\
-- Introduction to parabolic implosion in complex dimension 1.
-- Statement of the main result and proof.
-References
+- 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{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.
+- 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{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.
+- 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\\
-\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
+**Prerequisites:**\\
+- Basic complex analysis in dimension 1\\
-\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.
+- The Riemann sphere: its complex structures and the description of its group of holomorphic automorphisms\\
+- Basic discrete holomorphic dynamics on the Riemann sphere: local theory\\
-\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.
+- 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)**\\
+-Carne, Geometry and Groups https://www.dpmms.cam.ac.uk/~tkc/GeometryandGroups/GeometryandGroups.pdf\\
+-Carleson and Gamelin, Complex Dynamics, https://zr9558.files.wordpress.com/2013/11/complex-dynamics-carleson.pdf\\
+-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
 <WRAP center round box 60%>
@@ Line 36: / Line 40: @@
 <WRAP center round box 60%>
-[[ebsd2021:raissi2|Tema 2]]
+[[ebsd2021:raissi2|Mapas racionais na esfera de Riemann]]
 </WRAP>
 <WRAP center round box 60%>
-[[ebsd2021:raissi3|Tema 3]]
+[[ebsd2021:raissi3|Teoria local: dinâmica perto de pontos periódicos]]
 </WRAP>
 <WRAP center round box 60%>
-[[ebsd2021:raissi4|Tema 4]]
+[[ebsd2021:raissi4|Teoria global: conjuntos de Julia e de Fatou e suas propriedades]]
+</WRAP>
+<WRAP center round box 60%>
+[[ebsd2021:raissy5|Wandering domains for polynomials in higher dimension]]
 </WRAP>