Biaccessibility in quadratic Julia sets

This paper consists of two nearly independent parts, both of which discuss the common theme of biaccessible points in the Julia set J of a quadratic polynomial f : z bar arrow pointing right z(2) + c. In Part I, we assume that J is locally-connected. We prove that the Brolin measure of the set of biaccessible points (through the basin of attraction of infinity) in J is zero except when f (z) = z(2) - 2 is the Chebyshev map for which the corresponding measure is one. As a corollary, we show that a locally-connected quadratic Julia set is not a countable union of embedded arcs unless it is a straight line or a Jordan curve. In Part II, we assume that f has an irrationally indifferent fixed point alpha. If z is a biaccessible point in J, we prove that the orbit of z eventually hits the critical point of f in the Siegel case, and the fixed point a, in the Cremer case. As a corollary, it follows that the set of biaccessible points in J has Brolin measure zero.