ebsd2021:raissy
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| ebsd2021:raissy [2021/09/10 13:46] – escola | ebsd2021:raissy [2021/10/25 08:07] (current) – tahzibi | ||
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| - | ====== Wandering domains for polynomials in higher dimension ====== | ||
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| - | **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. | ||
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| - | **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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| 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 |
| - | + | 1-Carne, Geometry and Groups https:// | |
| + | 2-Carleson and Gamelin, Complex Dynamics, https:// | ||
| + | 3-Milnor, Dynamics in one complex variable, https:// | ||
| + | **Referencias para pre-requisitos (Português)**\\ | ||
| + | Lomonaco, Notas de aulas, https:// | ||
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ebsd2021/raissy.1631292412.txt.gz · Last modified: 2021/09/10 13:46 by escola