Vol. 14, No. 3, 2021

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Critical points of iterates of complex quadratic functions

Julia Barnes and James Marshall

Vol. 14 (2021), No. 3, 511–518

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.

complex quadratic functions, contour plots
Mathematical Subject Classification
Primary: 30D05
Received: 23 November 2020
Revised: 15 March 2021
Accepted: 17 March 2021
Published: 17 July 2021

Communicated by Michael Dorff
Julia Barnes
Mathematics and Computer Science
Western Carolina University
Cullowhee, NC
United States
James Marshall
Mathematics and Computer Science
Western Carolina University
Cullowhee, NC
United States