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.