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.