Line Complexes in Geometric Function Theory and 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: Nikolai Prochorov “Line Complexes in Geometric Function Theory and Dynamics” Let f be a complex polynomial in one variable, viewed as a self-map of the complex plane C. Suppose that f has exactly two critical values, namely −1 and 1. Consider the segment [−1, 1] and its preimage under f, i.e., f −1 ([−1, 1]). This set forms a planar tree in C. Moreover, the isomorphism class of this tree as a planar graph determines the polynomial f almost uniquely, up to pre-composition with an affine map. Such trees are known as dessins d’enfants and serve as a starting point for a rich theory developed by Grothendieck, Belyi, and others. This combinatorial construction extends naturally to polynomials with more than two critical values. Analogous constructions also exist for large classes of entire or meromorphic functions on C. For instance, it applies to maps of Speiser class — that is maps with finitely many singular values, i.e., values where not all inverse branches can be defined; examples include the functions cos, exp, and tan. In this broader setting, analogous combinatorial objects are often referred to as line complexes. In this minicourse, I will discuss how this combinatorial approach can be used to analyze function-theoretic and dynamical properties of wide classes of entire functions. 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).