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Shaun Bullet | Queen Mary University of London - Inglaterra

Abstract

The analogies between iteration of holomorphic maps and action of Kleinian groups were first enumerated by Sullivan in the late '80 when, in the landmark paper where he proved the 'no wandering theorem for rational maps on the Riemann sphere' by adapting the argument of the Ahlfors finiteness theorem, he also gives the first version of what it is called Sullivan's dictionary: a dictionary between the iteration of holomorphic maps and the Kleinian group worlds. Both rational maps and finitely generated Kleinian groups can be regarded as particular cases of correspondences. In 1994, S. Bullett and C. Penrose introduced the 1-parameter family of (2:2) holomorphic correspondences Fa (see the project for the definition) and proved that for every real parameter in a suitable disc Fa is a mating between a quadratic polynomial and the modular group. They conjectured that the connectedness locus for this family is homeomorphic to the Mandelbrot set. There have been several attempts to prove this long-standing conjecture, with posive partial results, but important technical problems have always been an obstacle for solving it completely. Since the host researcher introduced in her PhD thesis the object which can overcome these technical problems, she and S. Bullett have been working on solving the conjecture. We are finalizing our first joint paper, where we prove that every Fa with a in the connectedness locus is a mating between the modular group and a member of the family of quadratic rational maps with a parabolic fixed point (at infinity) with multiplier 1. Also, we are introducing a dynamical theory for this family Fa, and in particular, we are proving a Yoccoz inequality for this family. This is material for a second joint paper. Finally, we are working on proving that the connectedness locus for this family is homeomorphic to the parabolic mandelbrot set M1, which is homeomorphic to the Mandelbrot set by a result of Petersen and Roesch (and this is material of our third joint paper). During the visit, we plan to finalizing and submit our second joint paper, and hopefully the third one. (AU)

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