BRICS 2021 — Venue SH | Plenary Talk by Prof. Luna Lomonaco
Channel: BRICS Mathematics Conference 2021
Date: 2021-12-07

Title: Mating quadratic maps with the modular group
Speaker: Luna Lomonaco, IMPA Brazil

Abstract: Holomorphic correspondences are polynomial relations P (z, w) = 0, which can be regarded as multi-valued self-maps of the Riemann sphere, this is implicit maps sending z to w. The iteration of such a multi-valued map generates a dynamical system on the Riemann sphere: dynamical system which generalises rational maps and finitely generated Kleinian groups. We consider a specific 1-(complex) parameter family of (2 : 2) correspondences F_a (introduced by S. Bullett and C. Penrose in 1994), which we describe dynamically. In particular, we show that for every parameter in a subset of the parameter plane called ’the connectedness locus’ and denoted by MΓ, this family behaves as rational maps on a subset of the Riemann sphere and as the modular group on the complement: in other words, these correspondences are mating between the modular group and rational maps (in the family Per_1(1)). Moreover, we develop for this family of correspondences a complete dynamical theory which parallels the Douady-Hubbard theory of quadratic polynomials, and we show that MΓ is homeomorphic to the parabolic Mandelbrot set M1. This is joint work with S. Bullett (QMUL).