Specifying attracting cycles for Newton maps of polynomials

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 Newton map, or even the relaxed Newton map, for p has the given points as a super-attracting cycle. This improves the result in Plaza and Romero [6], which shows how to find such a polynomial of degree 2n. Moreover, we show that in general one cannot improve upon degree n+1. Our methods allow us to give a simple, constructive proof of the known result that for each cycle length n ≥ 2 and degree d ≥ 3, there exists a polynomial of degree d whose Newton map has a super-attracting cycle of length n.

Keywords:

• chaotic complex dynamics
• extraneous attractors
• Newton method
• relaxed Newton method
• polynomials
• rational maps

2000 Mathematics Subject Classification::

• 30D05
• 34M03
• 37F10
• 39B12

Acknowledgement

The authors would like to thank Paul Balister for helpful discussion regarding the proof of Theorem 1.3.

Notes

1. [email protected]

Additional information

Notes on contributors

Jared T. Collins

1