Simultaneously preperiodic integers for quadratic polynomials

Abstract

In this article, we study the set of parameters $c\in \mathbb{C}$ for which two given complex numbers $a$ and $b$ are simultaneously preperiodic for the quadratic polynomial $f_c(z)=z^2+c$. Combining complex-analytic and arithmetic arguments, Baker and DeMarco showed that this set of parameters is infinite if and only if $a^2=b^2$. Recently, Buff answered a question of theirs, proving that the set of parameters $c\in \mathbb{C}$ for which both $0$ and $1$ are preperiodic for $f_c$ is equal to $\{-2,-1,0\}$. Following his approach, we complete the description of these sets when $a$ and $b$ are two given integers with $|a|\ne |b|$.

13 pages, 6 figures