This note provides the complete factorization of the Misiurewicz-Thurston polynomial $q_{ \ell,n}=p_{ \ell+n}(z) - p_\ell(z)$ over $\mathbb{ C}$, which plays a central role in the study of the Mandelbrot set, where \[ p_0(z) = 0, \qquad p_{ n+1}(z) = p_n(z)^2 + z. \] The roots can be classified into two categories. First, there are hyperbolic points $\operatorname{ hyp}(k)$ for any divisor $k$ of $n$, which are parameters whose critical orbits are of exact period $k$. Those are roots of $q_{ \ell,n}$ with multiplicity $\left\lfloor \frac{ \ell -1}{ k} \right\rfloor + 2$. Next are the points $\operatorname{ mis}(j,k)$ for $2\leq j\leq \ell$ whose critical orbits are pre-periodic of exact period $k$ with an exact pre-period $j$. Those are simple roots of $q_{ \ell,n}$.
Categories: math.DS