Submitted by math_admin on Sun, 03/01/2020 - 20:47
preprint-id:
preprint-title:
Biaccessiblility in Quadratic Julia Sets II: The Siegel and Cremer Cases
preprint-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:
- In the Siegel case, the orbit of $z$ must eventually hit the critical point of $f$.
- 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.
preprint-year:
1998