New York Journal of Mathematics
Volume 29 (2023), 441-466


Devin Becker, Joanna Furno, and Lorelei Koss

Dynamical convergence of polynomials to products of power maps and the exponential

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Published: March 23, 2023.
Keywords: Complex dynamics, Julia sets, rational maps, entire functions.
Subject [2010]: 37F10, 37F46, 30D05.

We study the dynamics of a family of polynomial functions and the relationship to the dynamics of a related entire transcendental family of functions. As the degree of the polynomial approaches infinity, the polynomial functions converge uniformly on compact sets to a function that is a product of a power map and the exponential function. The advantage to the approach in the paper is that we can use the relatively simple, although high degree, polynomial functions to aid our understanding of the dynamics of the related transcendental entire function. We study properties both in the dynamical plane as well as in the parameter plane.


The first author acknowledges the financial support of an Undergraduate Research Assistantship Program (URAP) grant from DePaul University's College of Science and Health. The second author acknowledges the support of the National Science Foundation under Grant No. 1440140, while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the "Complex Dynamics: From Special Families to Natural Generalizations in One and Several Variables" semester of 2022.

Author information

Devin Becker
Department of Physics and MIT Kavli Institute
77 Massachusetts Ave, Cambridge, MA 02139, USA


Joanna Furno
Department of Mathematics and Statistics
University of South Alabama
411 University Blvd North, Mobile, AL 36688, USA


Lorelei Koss
Department of Mathematics and Computer Science
Dickinson College
P.O. Box 1773, Carlisle, PA 17013, USA;
Current address: Department of Mathematics
Colby College
5845 Mayflower Hill, Waterville, ME 04901, USA