Preperiodic points for families of polynomials

TL;DR

This paper investigates the conditions under which two polynomial families simultaneously have infinitely many parameters where both are preperiodic for a given family of polynomials, exploring a complex dynamical systems problem.

Contribution

It provides new insights into the simultaneous preperiodicity of polynomial values within parameterized families, extending understanding in complex dynamics.

Findings

Identifies conditions for infinite simultaneous preperiodic points

Establishes links between polynomial families and dynamical behavior

Contributes to the theory of preperiodic points in complex dynamics

Abstract

Let $a(\lambda)$ and $b(\lambda)$ be two polynomials with coefficients in complex numbers and let $f_{\lamb$ be a one-parameter family of polynomials indexed by all complex numbers $\lambda$. We study whether there exist infinitely many complex numbers $\lambda$ such that both $a(\lambda)$ and $b(\lambda)$ are preperiodic for $f_{\lambda}$.