Disconnected Julia Set Of Halley'S Method For Exponential Maps

We investigate the Halley method of exponential maps. Our main result is that, unlike Newton's method, the Julia set of Halley's method may be disconnected when applied to entire maps of form F=peq where p and q are polynomials and q is non-constant. We also describe the nature of the fixed points and classify rational Halley's maps of entire functions.