Iteration of the Translated Tangent

Abstract

The iteration of the map $$z \mapsto \lambda + \tan z$$ on the complex plane is investigated for each complex number $$\lambda $$ . It is shown that its Fatou set always contains a completely invariant component and the Julia set is either totally disconnected or connected. A necessary and sufficient condition for totally disconnected Julia set is proved. When the Julia set is connected, it is seen that the Fatou set can consist of additional components, which can be completely invariant attracting domain, a Siegel disk, a two periodic attracting domain or parabolic domain. Examples of each case are given. Further, a necessary and sufficient condition is provided for the Julia set to be a Jordan curve passing through infinity. It is proved that the set of parameters corresponding to Jordan curve Julia sets is connected, whereas the set of those corresponding to totally disconnected Julia sets is disconnected.