Dynamical convergence of a certain polynomial family to f_a(z) = z + e^z + a

Abstract

A transcendental entire function fa(z) = z +e z +a may have a Baker domain or a wandering domain, which never appear in the dynamics of polynomials. We consider a sequence of polynomials Pa,d(z) = (1+a/d)z +(1+z/d) d+1 +a, which converges uniformly on compact sets to fa as d → ∞. We show its dynamical convergence under a certain assumption, even though fa has a Baker domain or a wandering domain. We also investigate the parameter spaces of fa and Pa,d.