Constructing attracting cycles for Halley and Schröder maps of polynomials

Jared T. Collins

doi:10.3934/dcds.2017237

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2017, Volume 37, Issue 10: 5455-5465. Doi: 10.3934/dcds.2017237

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Constructing attracting cycles for Halley and Schröder maps of polynomials

Jared T. Collins^,

Department of Mathematics and Computer Science, Freed-Hardeman University, Henderson, TN 38340, USA

* Corresponding author: Jared T. Collins
* Corresponding author: Jared T. Collins

Received: September 2016

Revised: June 2017

Published: October 2017

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Abstract

We show that for any set of $n$ distinct points in the complex plane, there exists a polynomial $p$ of degree at most $n+1$ so that the corresponding Halley and Schröder map for $p$ has the given points as a super-attracting cycle. This improves the result in [1], which shows how to find such a polynomial of degree $3n$. Moreover we show that in general one cannot improve upon degree $n+1$.

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Mathematics Subject Classification: Primary: 30D05; Secondary: 34M03, 37F10, 39B12.

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References

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