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Cubic Polynomials with Periodic Cycles of a Specified Multiplier

Published online by Cambridge University Press:  20 November 2018

Patrick Ingram
Patrick Ingram*
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, ON and Department of Mathematics, Colorado State University, Fort Collins, CO, USA email: [email protected], [email protected]
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Abstract

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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 ${{\cup }_{n\ge 1}}\,\mathbb{C}\left( {{\lambda }^{1/n}} \right)$.

Keywords

37P35cubic polynomialsperiodic pointsholomorphic dynamics
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 64 , Issue 2 , 16 April 2012 , pp. 318 - 344
Copyright
Copyright © Canadian Mathematical Society 2012

References

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