A landing theorem for periodic rays of exponential maps

Abstract

For the family of exponential maps z ↦ exp ⁡ ( z ) + κ z\mapsto \exp (z)+\kappa , we show the following analog of a theorem of Douady and Hubbard concerning polynomials. Suppose that g g is a periodic dynamic ray of an exponential map with nonescaping singular value. Then g g lands at a repelling or parabolic periodic point. We also show that there are periodic dynamic rays landing at all periodic points of such an exponential map, with the exception of at most one periodic orbit.