Dynamical convergence of polynomials to the exponential

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Abstract
In this paper we investigate the relationship between the dynamics of the polynomials maps pd,λ(z)=(1+z/d)d and the exponential family Eλ(z)=λcz. We show that the hyperbolic components of the parameter 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 correspondings external rays for the polynomial families. For parameter values on the hairs, the Julia sets for the corresponding exponentials are the entire plane whereas, for polynomial parameters on the external rays, the Julia sets are Cantor sets

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