Factorization of the quadratic Misiurewicz-Thurston polynomials

Abstract

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 $\mathrm{hyp}(k)$ 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 $\lfloor \frac{\ell -1}{k}\rfloor +2$. Next are the points $\mathrm{mis}(j,k)$ for $2\le j\le \ell$ whose critical orbits are pre-periodic of exact period $k$ with an exact pre-period $j$. Those are simple roots of $q_{\ell ,n}$.

8 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:2402.06083