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^*= \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.