A parameter $c_0\in C$ 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 $c=c_0$. Information about the location of critical points and critical values of the multipliers might play a role in the study of the geometry of the Mandelbrot set.

It was previously shown by T. Firsova and the speaker that critical points of the multipliers equidistribute on the boundary of the Mandelbrot set $\mathbb M$, as $n\to\infty$. At the same time it was shown that the accumulation set $\mathcal X$ of the critical points of the multipliers is strictly larger than the boundary of $\mathbb M$, which is the support of the equilibrium measure. In this talk we will discuss further geometric properties of the accumulation set $\mathcal X$. In particular, we will show that it is bounded, path connected and contains the Mandelbrot set as a proper subset. This is joint work with Tanya Firsova.