Journal of Fractal Geometry

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Volume 6, Issue 1, 2019, pp. 89–108
DOI: 10.4171/JFG/70

Published online: 2019-02-01

Rational families converging to a family of exponential maps

Joanna Furno[1], Jane Hawkins[2] and Lorelei Koss[3]

(1) University of Houston, USA
(2) University of North Carolina at Chapel Hill, USA
(3) Dickinson College, Carlisle, USA

We analyze the dynamics of a sequence of families of non-polynomial rational maps, $\{f_{a,d}\}$, for $a \in \mathbb C^*= \mathbb C \setminus \{0\}, d \geq 2$. For each $d$, $\{f_{a,d}\}$ is a family of rational maps of degree $d$ of the Riemann sphere parametrized by $a \in \mathbb C^*$. For each $a \in \mathbb C^*$, as $d \to \infty$, $f_{a,d}$ converges uniformly on compact sets to a map $f_a$ that is conformally conjugate to a transcendental entire map on $\mathbb C$. We study how properties of the families $f_{a,d}$ contribute to our understanding of the dynamical properties of the limiting family of maps. We show all families have a common connectivity locus; moreover the rational maps contain some well-studied examples.

Keywords: Complex dynamics, Julia sets, rational maps, entire functions

Furno Joanna, Hawkins Jane, Koss Lorelei: Rational families converging to a family of exponential maps. J. Fractal Geom. 6 (2019), 89-108. doi: 10.4171/JFG/70