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 period n multiplier if map $f_{c_{0}}$ has periodic orbit , whose multiplier, viewed as locally analytic function c vanishing derivative at $c=c_{0}$ . We study accumulation set ${\mathcal X}$ points multipliers $n\to \infty $ This complements equidistribution result for that was previously obtained by authors. In particular, current paper, we prove bounded, connected, and contains Mandelbrot proper subset. also provide necessary sufficient condition outside to be contained show this satisfied an open parameters. Our similar flavor one conditions define set. As application, get sends Hausdorff dimension $f_{c}$ does not have