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2021 Critical points of iterates of complex quadratic functions
Julia Barnes, James Marshall
Involve 14(3): 511-518 (2021). DOI: 10.2140/involve.2021.14.511

Abstract

We look at a familiar one-parameter family of quadratic functions on the complex plane. After restricting the parameter to be real, we explore when the critical points of the functions and their iterates are real and when they are not real. We prove that when the parameter is greater than or equal to 2, all critical points are real. When the parameter is between 0 and 2, critical points for the original function are real but there is an iterate with nonreal critical points. When the parameter is equal to 0, all critical points are 0. When the parameter is less than 0, the critical points are all 0 or are nonreal. Finally, we compare the locations of these critical points to the contour plots of the real parts of the functions for different values of the parameter and for different iterates of the function.

Citation

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Julia Barnes. James Marshall. "Critical points of iterates of complex quadratic functions." Involve 14 (3) 511 - 518, 2021. https://doi.org/10.2140/involve.2021.14.511

Information

Received: 23 November 2020; Accepted: 17 March 2021; Published: 2021
First available in Project Euclid: 30 July 2021

Digital Object Identifier: 10.2140/involve.2021.14.511

Subjects:
Primary: 30D05

Keywords: complex quadratic functions , contour plots

Rights: Copyright © 2021 Mathematical Sciences Publishers

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Vol.14 • No. 3 • 2021
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