Dynamical convergence of polynomials to the exponential

In this paper we investigate the relationship between the dynamics of the p olynomial maps P-d,P-lambda(z)=(1 + z/d)(d) and the exponential family E-la mbda(z)= lambda e(z). We show that the hyperbolic components of the paramet er planes for the polynomials converge to those for the exponential family as the degree d tends to infinity. We also show that certain "hairs" in the parameter plane for the exponential are limits of corresponding external r ays for the polynomial families. For parameter values on the hairs, the jul ia sets for the corresponding exponentials are the entire plane whereas, fo r polynomial parameters on the external rays, the Julia sets are Canter set s. AMS Subject Classification: 58F23, 30D05.