Mating quadratic maps with the modular group |
| Luna Lomonaco |
| IMPA (Brasil) |
| 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 \[M_{\Gamma}\] , 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 \[{\rm Per}_1(1)\] ). 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 \[M_{\Gamma}\] is homeomorphic to the parabolic Mandelbrot set \[M_1\] . This is joint work with S. Bullett (QMUL). |