Mating quadratic maps with the modular group

Luna Lomonaco

Holomorphic correspondences are polynomial relations P(z,w)=0, which can be regarded as multi-valued self-maps of the Riemann sphere (implicit maps sending z to w). The iteration of such multi-valued map generates a dynamical system on the Riemann sphere (dynamical system which generalise 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 a in the connectedness locus M_{Γ}, this family is a mating between the modular group and rational maps in the family Per_1(1); we develop for this family a complete dynamical theory which parallels the Douady-Hubbard theory of quadratic polynomials; and we show that M_{Γ} is homeomorphic to the parabolic Mandelbrot set M_1.

http://scgp.stonybrook.edu/video_portal/video.php?id=4601

Simons- Program: Renormalization and universality in Conformal Geometry, Dynamics, Random Processes, and Field Theory