Bifurcation to an entire function

We will study bifurcations that occur in the topology of the Julia set of meromorphic functions as they become entire. We will characterize the Julia sets for special kinds of meromorphic maps and use well known results about the topology of the Julia set of the entire exponential function Elambda(z) = lambdae z.;In the dynamical plane of the stable exponentials the Julia set consists of closed Jordan arcs, called "hairs", which have the property that all points on the hair share the same itinerary, given some suitable partitioning of the plane. In the case of Elambda (z) = lambdaez where lambda ∈ (0,1/e) it is possible to parameterize all hairs in the Julia set, showing that in this case the Julia set is a Cantor bouquet. Only the endpoints of the hairs are accessible from the basin of attraction. This can be shown by uniformizing the basin of attraction and taking the limit along a ray from the origin to some point of the unit disk.;We will see that in fact the family of all maps with constant Schwarzian derivative is a two parameter family that includes the exponential and the tangent family. We will characterize the Julia set of maps that have exactly one attracting fixed point, two attracting fixed points and an attracting two-cycle. For the case where the map has exactly one attracting fixed point, we will show that the Julia set is a Cantor set. The Julia set for the map that has two attracting fixed points is a Jordan curve going through infinity.;Within these two sets of functions we will approach the entire function and we will witness the Julia set changing from a Cantor set to a Cantor bouquet as we approach the entire function with exactly one attracting fixed point from the meromorphic maps that have exactly one attracting fixed point. As we approach the same exponential map from the meromorphic map with two attracting fixed points we will witness the Julia set changing from a Jordan curve to the Cantor Bouquet.