Institute for Mathematical Sciences
Biaccessiblility in Quadratic Julia Sets II: The Siegel and Cremer Cases.
Abstract: Let $f$ be a quadratic polynomial which has an irrationally indifferent fixed point $\alpha$. Let $z$ be a biaccessible
point in the Julia set of $f$. Then:
1. In the Siegel case, the orbit of $z$ must eventually hit
the critical point of $f$.
2. In the Cremer case, the orbit of $z$ must eventually hit
the fixed point $\alpha$.
Siegel polynomials with biaccessible critical point certainly
exist, but in the Cremer case it is possible that biaccessible
points can never exist.
As a corollary, we conclude that the set of biaccessible points
in the Julia set of a Siegel or Cremer quadratic polynomial has
Brolin measure zero.
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