Fine structure of connectedness loci

Abstract

We study distributions of the harmonic and Lebesgue measures on the boundary of the connectedness locus $\mathcal{M}_{d}$ for uni-critical polynomials $z^{d}+c$ and dynamical properties which are typical with respect to the harmonic measure. One result is a conformal similarity between $\mathcal{M}_{d}$ and the corresponding Julia set for almost every point of $\mathcal{M}_{d}$ with respect to the harmonic measure. It is also shown that the harmonic measure is supported on Lebesgue density points of the complement of $\mathcal{M}_{d}$ which are not accessible from outside within John angles and at which the boundary of $\mathcal{M}_{d}$ ‘spirals’ infinitely often in both directions. In the case of quadratic polynomials, we prove that every parameter c from the boundary of the Mandelbrot set $\mathcal{M}$ that is recurrent under iterates of $z^{2}+c$ but not infinitely tunable is a Lebesgue weak density point of the complement of $\mathcal{M}$. A direct consequence of our theory is the Yoccoz local connectivity theorem. A new technical ingredient in the paper is the method of “amplification” which is based on promoting the small scale geometry of the parameter space at a typical point in the sense of harmonic measure to uniform geometry of the corresponding Julia set. The pointwise conformality of the similarity map is derived through inducing techniques and a classical TWB-theory.