Pure Maths Colloquium: Mandelbrot Sets for Matings
30 November 2015 16:00 in CM221
The classical Mandelbrot set M is the subset of parameter space for which the Julia set of the quadratic polynomial z^2 + c is connected. Two analogous connectivity loci are M(1) for the family of rational maps containing matings of z^2+c with z^2+1/4, and
M(corr) for the family of quadratic holomorphic correspondences containing matings between quadratic polynomials and the modular group PSL(2,Z). Computer plots have long suggested that these three “Mandelbrot Sets’” are homeomorphic to one another.
Carsten Petersen and Pascale Roesch have announced a proof that M(1) is homeomorphic to M (not yet published). Luna Lomonoco (Universidade de Sao Paulo) and I have a detailed strategy to prove that M(corr) is homeomorphic to M(1). I will discuss the notion of a “mating’’ between two holomorphic dynamical systems, and outline our proposed proof that M(corr) is homeomorphic to M(1), focusing in particular on a new inequality analogous to that proved by Yoccoz for quadratic polynomials, and introducing the methods involved, which range from hyperbolic and quasiconformal geometry to the symbolic dynamics of Sturmian sequences.
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