A survey on non-autonomous basins in several complex variables

Consider a holomorphic automorphism which acts hyperbolically on some invariant compact set. Then for every point in the compact set there exists a stable manifold, which is a complex manifold diffeomorphic to real Euclidean space. If the point is fixed, then the stable manifold is even biholomorphic to complex Euclidean space. In fact, it is known that the stable manifold of a generic point is biholomorphic to Euclidean space, and it has been conjectured that this holds for every point. In this article we survey the history of this problem, addressing both known results and the techniques used to obtain those results. Moreover, we present a list of seemingly simpler open problems and prove several new results, all pointing towards a positive answer to the conjecture discussed above.