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Teoria Ergódica

Satellite copies of the Mandelbrot set
Luna Lomonaco

Quinta-feira, 11 de outubro de 2018, 15:30
Sala 228

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.