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# Cubic Polynomials with Periodic Cycles of a Specified Multiplier

Published:2012-03-05
Printed: Apr 2012
• Patrick Ingram,
Department of Pure Mathematics, University of Waterloo
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## 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 $\bigcup_{n\geq 1}\mathbb{C}(\lambda^{1/n})$.
 Keywords: cubic polynomials, periodic points, holomorphic dynamics
 MSC Classifications: 37P35 - Arithmetic properties of periodic points