Cubic Polynomials with Periodic Cycles of a Specified Multiplier

http://dx.doi.org/10.4153/CJM-2011-093-8
Canad. J. Math. 64(2012), 318-342

  Published:2012-03-05
 Printed: Apr 2012

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 $\bigcup_{n\geq 1}\mathbb{C}(\lambda^{1/n})$.