Neutral Phenomenon in Complex Dynamics (Lecture 3)

Continued Fractions in Fractals, Ergodic theory and Dynamics Thematic week Holomorphic Dynamics and related fields, Warsaw, 11 – 15 May 2026 Lecture by: Willie Rush Lim “Neutral Phenomenon in Complex Dynamics” This mini-course will be a survey of fundamental problems and classical and modern results of neutral complex dynamics. That is, the dynamics of a holomorphic map of one complex variable with a neutral fixed point; the simplest family of examples is quadratic polynomials ft(z) = e 2πitz + z 2. Within this simple formula lies a rich theory. Historically, the development relies heavily on the Diophantine properties of the rotation number t. The goal of this mini-course is two-fold: a survey of classical results and an overview of advances in the last 5 years. I will discuss the following topics. Linearization problem: A classical result by Brjuno and Yoccoz states that such a map is analytically linearizable near the neutral fixed point if the rotation number t is a Brjuno irrational. I will explain Yoccoz’s proof in the framework of sector renormalization. Hedgehogs and circle maps: There is a strong connection between attractors of maps with neutral fixed points with the study of circle diffeomorphisms and critical circle maps. I will discuss this connection, following the works of Perez-Marco, Herman, Douady, Ghys, etc. Golden mean case: This is one of the most well understood cases. There is a complete understanding of the topology of the Julia set of ft (Petersen). Even more surprising, there is also a remarkable rigidity, self-similarity, and universality phenomenon (McMullen). All irrationals: I will talk about the most recent development that covers all irrationals t. The most recent breakthrough was by Dima Dudko and Misha Lyubich on uniform bounds for renormalization. I will talk about various consequences of these bounds on the properties of the full attractor (Mother Hedgehog) of ft and a generalization of dynamical universality. This lecture was partially supported by the Simons Foundation grant (award no. SFI-MPS-T-Institutes-00010825) and from State Treasury funds as part of a task commissioned by the Minister of Science and Higher Education under the project “Organization of the Simons Semesters at the Banach Center - New Energies in 2026-2028” (agreement no. MNiSW/2025/DAP/491).