Critical points of the multiplier map for the quadratic family

Abstract

The multiplier ${\lambda }_n$ of a periodic orbit of period $n$ can be viewed as a (multiple-valued) algebraic function on the space of all complex quadratic polynomials $p_c(z)=z^2+c$. We provide a numerical algorithm for computing critical points of this function (i.e., points where the derivative of the multiplier with respect to the complex parameter $c$ vanishes). We use this algorithm to compute critical points of ${\lambda }_n$ up to period $n=10$.