Typical orbits of quadratic polynomials with a neutral fixed point I: non-Brjuno type

Details Typical orbits of quadratic polynomials with a neutral fixed point I: non-Brjuno type Typical orbits of quadratic polynomials with a neutral fixed point I: non-Brjuno type

2010-01-22

arxiv.org/abs/1001.4030v1

Mathematics

Dynamical Systems

47 pages, 5 figures,

Scientific paper

We study (Lebesgue) typical orbits of quadratic polynomials $P_a(z)=e^{2\pi a} z+z^2: C -> C$, with $a$ of non-Brjuno and high return type. This includes quadratic polynomials with positive area Julia set of X. Buff and A. Cheratat. As a consequence, we introduce rational maps of arbitrarily large degree for which the Brjuno condition is optimal for their linearizability. Our technique uses the near-parabolic renormalization developed by H. Inou and M. Shishikura.

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