Typical orbits of quadratic polynomials with a neutral fixed point II: Brjuno type

Details Typical orbits of quadratic polynomials with a neutral fixed point II: Brjuno type Typical orbits of quadratic polynomials with a neutral fixed point II: Brjuno type

2012-02-10

arxiv.org/abs/1202.2282v2

Mathematics

Dynamical Systems

30 pages, 5 figures

Scientific paper

We describe the topological behavior of typical orbits of complex quadratic polynomials $P_\alpha(z)=e^{2\pi i \alpha} z+z^2$, with $\alpha$ of high return type. Here we prove that for such Brjuno values of $\alpha$ the closure of the critical orbit, which is the measure theoretic attractor of the map, has zero area. Then combining with Part I of this work, we show that the limit set of the orbit of a typical point in the Julia set is equal to the closure of the critical orbit.

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