Dynamical approximation of postsingularly finite exponentials

Abstract

Given any postsingularly finite exponential function $p_{\lambda }(z)=\lambda \exp (z)$ where $λ\in \C^*$, we construct a sequence of postcritically finite unicritical polynomials $p_{d,{\lambda }_d}(z)={\lambda }_d(1+\frac{z}{d}{)}^d$ that converge to $p_{\lambda }$ locally uniformly in $\C$, with the same postsingular portrait as that of $p_{\lambda }$. We describe ${\lambda }_d$ in terms of parameter rays in the space of degree $d$ unicritical polynomials, and exhibit a relationship between the angles of these parameter rays as $d\to \infty$ and the external addresses associated with $\lambda$ in the exponential parameter plane.

42 pages, 7 figures