Joining polynomial and exponential combinatorics for some entire maps

Abstract

We consider families of entire transcendental maps given by F λ,m (z) = λz m exp(z) where m ≥ 2. All these maps have a superattracting fixed point at z = 0 and a free critical point at z = −m. In parameter planes we focus on the capture zones, i.e., we consider λ values for which the free critical point belongs to the basin of attraction of z = 0. We explain the connection between the dynamics near zero and the dynamics near infinity at the boundary of the immediate basin of attraction of the origin, thus, joining together exponential and polynomial behaviors in the same dynamical plane.

Rational families converging to a family of exponential maps

“…The usefulness of this approach is that lower degree families have been studied earlier ( [6,12]), and all families have a common connectivity locus (Theorem 2.1 below). Other authors have published results about polynomial approximations to entire maps (see e.g., [2,4,5]); we show that with one exception in degree 2 our maps are not polynomial maps.…”