Accumulation Theorems for Quadratic Polynomials

We consider the one-parameter family of quadratic polynomials: P_t(z) = z² + c_t, where c_t = e^2πit/2(1 - e2πit/2), t ∈ double-struck T sign : = ℝ/ℤ, i.e. monic centered quadratic polynomials with an indifferent fixed point αt and prefixed point -α_t. Let A_t be any one of the sets {0, ±α_t}, {±α;_t}, {0, α_t}, or {0, -α_t}. Then we prove that for quadratic Julia sets corresponding to a G_δ-dense subset of double-struck T sign there is an explicitly given external ray accumulating on At. In the case At = {±α_t} the theorem is known as the Douady accumulation theorem. Corollaries are the non-local connectivity of these Julia sets and the fact that all such Julia sets contain a Cremer point. Existence of non-locally connected quadratic Julia sets of Hausdorff dimension two is derived by using a recent result of Shishikura. By tuning, the results hold on the boundary of any hyperbolic component of the Mandelbrot set. Finally, we concentrate on quadratic Cremer point polynomials. Here we prove that any ray accumulating on two symmetrical points of the Julia set must accumulate the origin. As a consequence, the dense G_δ-sets arising from the first two possible choices of At are the same. We also prove that, if two distinct rays accumulate both to two distinct points, then the rays must accumulate on a common continuum joining the two points. This supports the conjecture that αt and -α_t may be joined by an arc in the Julia set.