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Biaccessiblility in quadratic Julia sets II: The Siegel and Cremer cases

Saeed Zakeri
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Abstract

Let $f$ be a quadratic polynomial which has an irrationally indifferent fixed point $α$. 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 $α$. 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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Saeed Zakeri. 1998-01-15. Biaccessiblility in quadratic Julia sets II: The Siegel and Cremer cases. https://arxiv.org/abs/math/9801150

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