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Published online by Cambridge University Press: 20 November 2018
We consider cubic polynomials $f\left( z \right)\,=\,{{z}^{3}}\,+\,az\,+\,b$ defined over $\mathbb{C}\left( \lambda \right)$, 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 $N\,\ge \,3$, 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 ${{\cup }_{n\ge 1}}\,\mathbb{C}\left( {{\lambda }^{1/n}} \right)$.