The secant map applied to a real polynomial with multiple roots

Antonio Garijo; Xavier Jarque

doi:10.3934/dcds.2020133

Article Contents

2020, Volume 40, Issue 12: 6783-6794. Doi: 10.3934/dcds.2020133

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The secant map applied to a real polynomial with multiple roots

Antonio Garijo^1, , and
Xavier Jarque^2,

1.
Departament d'Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, 43007 Tarragona, Catalonia
2.
Departament de Matemàtiques i Informàtica, Universitat de Barcelona, 08007 Barcelona, Catalonia

^* Corresponding author: antonio.garijo@urv.cat
^* Corresponding author: antonio.garijo@urv.cat

Received: July 2019

Revised: September 2019

Early access: February 2020

Published: August 2020

This work has been partially supported by MINECO-AEI grants MTM-2017-86795-C3-2-P and MTM-2017-86795-C3-3-P, the Maria de Maeztu Excellence Grant MDM-2014-0445 of the BGSMath and the AGAUR grant 2017 SGR 1374

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Abstract

We investigate the plane dynamical system given by the secant map applied to a polynomial $ p $ having at least one multiple root of multiplicity $ d>1 $. We prove that the local dynamics around the fixed points related to the roots of $ p $ depend on the parity of $ d $.

Keywords:

Mathematics Subject Classification: Primary: 37G35, 37N30; Secondary: 37C70.

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Figure 1. Dynamical plane of the secant map applied to $ p(x) = (x+2)x(x-1)^d $ for several values of $ d $. We show in red (dark grey) the basin of attraction of the multiple root of $ p $ corresponding to the fixed point of the secant map located at $ (1,1) $, in green (light grey) the basin of attraction of $ (-2,-2) $ and in blue (black) the basin of attraction of $ (0,0) $. The white regions that appear in each of the pictures are in the basin of a critical point of $ p $. The range of the pictures (a), (c), (e) and (f) is [-3, 3]x[-3, 3]

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Figure 2. Dynamics of T near a simple focal point Q

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References

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