Rational families converging to a family of exponential maps

Abstract

We analyze the dynamics of a sequence of families of non-polynomial rational maps, $\{f_{a,d}\}$, for $a\in {\mathbb{C}}^{\ast }=\mathbb{C}\setminus \{0\},d\ge 2$. For each $d$, $\{f_{a,d}\}$ is a family of rational maps of degree $d$ of the Riemann sphere parametrized by $a\in {\mathbb{C}}^{\ast }$. For each $a\in {\mathbb{C}}^{\ast }$, 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.