Abstract view
Cubic Polynomials with Periodic Cycles of a Specified Multiplier


Published:20120305
Printed: Apr 2012
Patrick Ingram,
Department of Pure Mathematics, University of Waterloo
Abstract
We consider cubic polynomials $f(z)=z^3+az+b$ defined over
$\mathbb{C}(\lambda)$, with a marked point of period $N$ and multiplier
$\lambda$. In the case $N=1$, there are infinitely many such objects,
and in the case $N\geq 3$, only finitely many (subject to a mild
assumption). The case $N=2$ has particularly rich structure, and we
are able to describe all such cubic polynomials defined over the field
$\bigcup_{n\geq 1}\mathbb{C}(\lambda^{1/n})$.