The Mandelbrot set and its Satellite copies

Abstract: 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 proved the existence of homeomorphic copies of M inside of M, which can be primitive (roughly speaking the ones with a cusp) or a satellite (without a cusp). Lyubich proved that the primitive copies of M are quasiconformally homeomorphic to M, and that the satellite ones are quasiconformally homeomorphic to M outside any small neighbourhood of the root. The satellite copies are not quasiconformally homeomorphic to M, but are they mutually quasiconformally homeomorphic? In a joint work with C. Petersen we prove that this question has in general a negative answer

PortugueseMathematics

Audience: general audience

Seminários de Matemática da UFPB