Fatou Component Classification and Wandering Domains in Holomorphic Dynamics
Iterating a holomorphic map splits its domain into the Fatou set, where the iterates form a normal family and dynamics are orderly, and the Julia set, where nearby orbits separate. A student learns the four possible types of an invariant Fatou component of a rational map on the Riemann sphere -- (super)attracting domain, parabolic domain, Siegel disk, Herman ring -- why every Fatou component in dimension one is pre-periodic, and how parabolic implosion and Lavaurs maps break that picture in complex dimension two, where polynomial skew-products over a parabolic fiber admit a genuinely wandering Fatou component.
Introduction to Fatou components in holomorphic dynamics
Lecture notes on how iterating a holomorphic map splits its domain into an orderly part and a chaotic part. For a rational map f of degree at least 2 on the Riemann sphere, the Fatou set is the large…