Polynomial skew-products in dimension 2: bulging and wandering Fatou components (Q1680755)

Let \(f: {\mathbb P}_1\to{\mathbb P}_1\) be a rational map of degree \(d>1\) on the Riemann sphere. Then the classification of an invariant Fatou component \(\Omega\) of \(f\) is well understood. In particular a result of Sullivan states that every Fatou component of \(f\) is non-wandering. The article is expository in respect to this result, namely it expounds extensively the research directions as well as vast references on the current efforts to understand this result in different contexts for various maps \(f\) in higher dimensions.