Accumulation set of critical points of the multipliers in the quadratic family

Abstract

A parameter $c_0\in \mathbb{C}$ in the family of quadratic polynomials $f_c(z)=z^2+c$ is a critical point of a period $n$ multiplier, if the map $f_{c_0}$ has a periodic orbit of period $n$, whose multiplier, viewed as a locally analytic function of $c$, has a vanishing derivative at $c=c_0$. We study the accumulation set $\mathcal{X}$ of the critical points of the multipliers, as $n\to \infty$. This study complements the equidistribution result for the critical points of the multipliers that was previously obtained by the authors. In particular, in the current paper we prove that the accumulation set $\mathcal{X}$ is bounded, path connected and contains the Mandelbrot set as a proper subset. We also provide a necessary and sufficient condition for a parameter outside of the Mandelbrot set to be contained in the accumulation set $\mathcal{X}$ and show that this condition is satisfied for an open set of parameters. Our condition is similar in flavor to one of the conditions that define the Mandelbrot set. As an application, we get that the function that sends $c$ to the Hausdorff dimension of $f_c$, does not have critical points outside of the accumulation set $\mathcal{X}$.

Corollary 1.3 about the relation to the Hausdorff dimension of the Julia sets is included