Simultaneously preperiodic points for families of polynomials in normal form

TL;DR

This paper proves that for a family of polynomials with specific parameters, the set of parameters making multiple points preperiodic is contained within finitely many hypersurfaces, revealing a finiteness property in complex dynamics.

Contribution

It establishes a finiteness result for parameters where multiple points are simultaneously preperiodic in a family of polynomials in normal form.

Findings

Preperiodic points are contained in finitely many hypersurfaces.

The result applies to polynomials with parameters in complex space.

It advances understanding of dynamical stability in polynomial families.

Abstract

Let $d>m>1$ be integers, let $c_1,\dots, c_{m+1}$ be distinct complex numbers, and let $\mathbf{f}(z):=z^d+t_1z^{m-1}+t_2z^{m-2}+\cdots + t_{m-1}z+t_m$ 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 $\mathbb{A}^m$.