Rational families converging to a family of exponential maps

Abstract

We analyze the dynamics of a sequence of families of non-polynomial rational maps, tf a,d u, for a P C˚" Czt0u, d ě 2. For each d, tf a,d u is a family of rational maps of degree d of the Riemann sphere parametrized by a P C˚. For each a P C˚, as d Ñ 8, f a,d converges uniformly on compact sets to a map fa that is conformally conjugate to a transcendental entire map on 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.