Mathematical Research Letters

Volume 16 (2009)

Number 1

Uniform Bounds on Pre-Images under Quadratic Dynamical Systems

Pages: 87 – 101



Xander Faber (McGill University)

Benjamin Hutz (Amherst College)

Patrick Ingram (University of Waterloo)

Rafe Jones (College of the Holy Cross)

Michelle Manes (University of Hawaii)

Thomas J. Tucker (University of Rochester)

Michael E. Zieve (Rutgers University)


For any elements $a,c$ of a number field $K$, let $\Gamma(a,c)$ denote the backwards orbit of $a$ under the map $f_c\colon\CC\to\CC$ given by $f_c(x)=x^2+c$. We prove an upper bound on the number of elements of $\Gamma(a,c)$ whose degree over $K$ is at most some constant $B$. This bound depends only on $a$, $[K:\QQ]$, and $B$, and is valid for all $a$ outside an explicit finite set. We also show that, for any fixed $N\ge 4$ and any $a\in K$ outside a finite set, there are only finitely many pairs $(y_0,c)\in\CC^2$ for which $[K(y_0,c)\col K]<2^{N-3}$ and the value of the $N\tth$ iterate of $f_c(x)$ at $x=y_0$ is $a$. Moreover, the bound $2^{N-3}$ in this result is optimal.

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