The Mandelbrot set and its satellite copies

Monday, February 27, 2017
Ora Inizio: 
Ora Fine: 
Luna Lomonaco
Universidade de São Paulo
For a polynomial on the Riemann sphere, infinity is a (super) attracting fixed point, and the filled Julia set is the set of points with bounded orbit. Consider the quadratic family $P_c(z)=z^2+c.$ The Mandelbrot set M  is the set of parameters c such that the filled Julia set of $P_c$ is connected. Douady and Hubbard, using polynomial-like mappings, proved the existence of homeomorphic copies of M inside of M, which can be primitive (if, roughly speaking, they have a cusp) or satellite (if they don't). They conjectured that the primitive copies of M are quasiconformal homeomorphic to M, and that the satellite ones are quasiconformal homeomorphic to M outside any small neighbourhood of the root. These results are now theorems due to Lyubich. The satellite copies are not quasiconformal homeomorphic to M, but are they mutually quasiconformally homeomorphic? In a joint work with C. Petersen we prove that this question, which has been open for about 20 years has in general a negative answer.