Biaccessiblility in Quadratic Julia Sets Ii the Siegel and Cremer Cases

Let f be a quadratic polynomial which has an irrationally indi erent xed point 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 xed point 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 x Introduction Let f be a polynomial map of the complex plane C A xed point z f z is called indi erent if the multiplier f z has the form e i where the rotation number belongs to R Z We call z irrationally indi erent if is irrational so that is on the unit circle but not a root of unity Let z be an irrationally indi erent xed point of f When f is holomorphically linearizable about z we call z a Siegel xed point On the other hand when z is nonlinearizable it is called a Cremer xed point In this paper we only consider quadratic polynomials Such a polynomial which we can put in the normal form