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October  2017, 37(10): 5455-5465. doi: 10.3934/dcds.2017237

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

Received  September 2016 Revised  June 2017 Published  June 2017

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 $n+1$.

Keywords: Complex dynamics, extraneous attractors, Halley's method, Schröder's method, polynomials.
Mathematics Subject Classification: Primary: 30D05; Secondary: 34M03, 37F10, 39B12.
Citation: Jared T. Collins. Constructing attracting cycles for Halley and Schröder maps of polynomials. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5455-5465. doi: 10.3934/dcds.2017237
References:
[1]

S. Amat, S. Busquier and S. Plaza, A construction of attracting periodic orbits for some classical third-order iterative methods, J. Comput. Appl. Math., 189 (2006), 22-33.  doi: 10.1016/j.cam.2005.03.049.  Google Scholar

[2]

J. Campbell and J. Collins, Specifying attracting cycles for Newton maps of polynomials, J. Difference Equ. Appl., 19 (2013), 1361-1379.  doi: 10.1080/10236198.2012.751987.  Google Scholar

[3]

D. Hilbert, Über die vollen Invariantensysteme, Math. Ann., 42 (1893), 313-373.  doi: 10.1007/BF01444162.  Google Scholar

[4]

K. Kneisl, Julia sets for the super-Newton method, Cauchy's method and Halley's method, Chaos, 11 (2001), 359-370.  doi: 10.1063/1.1368137.  Google Scholar

[5]

S. Plaza and N. Romero, Attracting cycles for the relaxed Newton's method, J. Comput. Appl. Math., 235 (2011), 3238-3244.  doi: 10.1016/j.cam.2011.01.010.  Google Scholar

[6]

S. Plaza and V. Vergara, Existence of attracting periodic orbits for the Newton method, Sci. Ser. A Math. Sci., 7 (2001), 31-36.   Google Scholar

show all references

References:
[1]

S. Amat, S. Busquier and S. Plaza, A construction of attracting periodic orbits for some classical third-order iterative methods, J. Comput. Appl. Math., 189 (2006), 22-33.  doi: 10.1016/j.cam.2005.03.049.  Google Scholar

[2]

J. Campbell and J. Collins, Specifying attracting cycles for Newton maps of polynomials, J. Difference Equ. Appl., 19 (2013), 1361-1379.  doi: 10.1080/10236198.2012.751987.  Google Scholar

[3]

D. Hilbert, Über die vollen Invariantensysteme, Math. Ann., 42 (1893), 313-373.  doi: 10.1007/BF01444162.  Google Scholar

[4]

K. Kneisl, Julia sets for the super-Newton method, Cauchy's method and Halley's method, Chaos, 11 (2001), 359-370.  doi: 10.1063/1.1368137.  Google Scholar

[5]

S. Plaza and N. Romero, Attracting cycles for the relaxed Newton's method, J. Comput. Appl. Math., 235 (2011), 3238-3244.  doi: 10.1016/j.cam.2011.01.010.  Google Scholar

[6]

S. Plaza and V. Vergara, Existence of attracting periodic orbits for the Newton method, Sci. Ser. A Math. Sci., 7 (2001), 31-36.   Google Scholar

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