Rational families converging to a family of exponential maps

Abstract

We analyze the dynamics of a sequence of families of non-polynomial rational maps, tfa,du, for a P C ̊ “ Czt0u, d ě 2. For each d, tfa,du 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, fa,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 fa,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. Mathematics Subject Classification (2010). 37F10, 37F45, 30D05.