Simultaneously preperiodic points for families of polynomials in normal form

Abstract

Let $d>m>1$ be integers, let $c_1,\dots ,c_{m+1}$ be distinct complex numbers, and let $\mathbf{f}(z):=z^d+t_1{z}^{m-1}+t_2{z}^{m-2}+\cdots +t_{m-1}z+t_m$ be an $m$-parameter family of polynomials. We prove that the set of $m$-tuples of parameters $(t_1,\dots ,t_m)\in {\mathbb{C}}^m$ with the property that each $c_i$ (for $i=1,\dots ,m+1$) is preperiodic under the action of the corresponding polynomial $\mathbf{f}(z)$ is contained in finitely many hypersurfaces of the parameter space ${\mathbb{A}}^m$.

9 pages