Dynamical Systems and Analysis Seminar - Lasse Rempe

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Speaker: Lasse Rempe (University of Manchester) Title: Absence of bounded-orbit wandering domains Abstract: Let f be an analytic self-map of either the complex plane or the Riemann sphere. We may consider f as describing the transition rule of a dynamical system, and study the behaviour of points under iteration (repeated application) of the function f. The Fatou set F(f) consists of those points near which the dynamics is stable under small perturbations of the starting value. A connected component of F(f) is called a Fatou component. A Fatou component is called wandering if its forward iterates are pairwise disjoint. A famous theorem of Sullivan from 1985 states that rational maps have no wandering domains. His proof used quasiconformal deformation theory, and relied crucially on the fact that rational maps of fixed degree depend only on finitely many parameters. Transcendental (i.e., non-polynomial) self-maps of the complex plane may have wandering domains. A long-standing open question asked whether it is possible for orbits in such a domain to remain bounded under iteration. We answer this question by proving that bounded-orbit wandering domains do not exist. (This is joint work with Drach, Pardo-Simón, Prochorov, U?akar and Waterman. Since our result was first announced, Ye also announced a proof of the main result.) This result was obtained with the help of generative AI, based on a new proof of Sullivan's theorem discovered by Ye, also using generative AI. We have also formalised the proof using the Lean proof assistant. I will discuss some thoughts on what the rapid advances in AI capabilities may mean for holomorphic dynamics (and mathematics) going forward. Room: Frank Adams 1 Further information: https://personalpages.manchester.ac.uk/staff/yotam.smilansky/dynamics_analysis