Critical points of the multiplier map for the quadratic family
This work provides computational tools and data for studying the dynamics of quadratic polynomials, but it is an incremental contribution within a specialized domain.
The authors developed a numerical algorithm to compute critical points of the multiplier map for quadratic polynomials and applied it up to period 10, providing the first such data for periods 9 and 10.
The multiplier $λ_n$ 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 $λ_n$ up to period $n=10$.