Title: Sullivan's Proof for the No Wandering Domains Theorem
Abstract:
Rational maps of the Riemann sphere do not admit wandering components in their Fatou sets. This result, conjectured by Fatou and Julia in the 1920s, was unusually elusive to prove, and was resolved by Sullivan in 1985. The proof, surprisingly, required the introduction of quasiconformal surgery into holomorphic dynamics, introducing a whole new tool box to the field.
We will review the statement of the theorem, and the proof's core novel mechanisms: using the Measurable Riemann Mapping Theorem to show that a wandering domain would allow for an infinite-dimensional space of deformations, contradicting the finite-dimensionality of the space of rational maps.