Mating quadratic maps with the modular group

Abstract:

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 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 a in the connectedness locus $M_{\Gamma}$, 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 $M_{\Gamma}$ is homeomorphic to the parabolic Mandelbrot set $M_1$. This is joint work with S. Bullett (QMUL)