New York Journal of Mathematics
Volume 16 (2010) 749-761


Julia A. Barnes, Clinton P. Curry, and Lisbeth E. Schaubroeck

Real and imaginary parts of polynomial iterates

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Published: December 15, 2010
Keywords: Filled Julia sets, complex dynamical systems, Böttcher uniformization
Subject: 37

Julia sets for complex-valued polynomials have been well-studied for years. However, the graphs of the polynomials themselves and their iterates are more difficult to visualize because they are four-dimensional. In this paper, we explore the dynamics of these functions by analyzing the behavior of the real and imaginary parts of the iterates. We also define two sets of points for which the real (respectively imaginary) parts of the iterates remain bounded, and prove how these sets relate to the corresponding filled Julia set. We end by applying our results to the well-known class of functions fc(z) =z2+c.

Author information

Julia A. Barnes:
Western Carolina University

Clinton P. Curry:
Stony Brook University

Lisbeth E. Schaubroeck:
United States Air Force Academy