Iteration of some topologically hyperbolic maps in the family $\lambda +z+\tan z$

Abstract

Iteration of the function $f_{\lambda }(z)=\lambda +z+\tan z,z\in \mathbb{C}$ is investigated in this article. It is proved that for every $\lambda$, the Fatou set of $f_{\lambda }$ has a completely invariant Baker domain $B$; we call it the primary Fatou component. The rest of the results deals with $f_{\lambda }$ when it is topologically hyperbolic. For all real $\lambda$ or $\lambda$ such that $\lambda =\pi k+i{\lambda }_2$ for some integer $k$ and $0<{\lambda }_2<1$, the only other Fatou component is shown to be another completely invariant Baker domain. It is proved that if $|2+{\lambda }^2|<1$, then the Fatou set is the union of $B$ and infinitely many invariant attracting domains. Every such domain $U$ has exactly one invariant access to infinity and is unbounded in a special way; $\{\Im (z):z\in U\}$ is unbounded whereas $\{\Re (z):z\in U\}$ is bounded. If $\Im (\lambda )>\sqrt{2}+{\sinh }^{-1}1$ then it is found that the primary Fatou component is the only Fatou component and the Julia set is disconnected. For every natural number $k$, the Fatou set of $f_{\lambda }$ for $\lambda =k\pi +i\frac{\pi }{2}$ is shown to contain $k$ wandering domains with distinct grand orbits. These wandering domains are found to be escaping. The Fatou set is the union of $B$, these wandering domains and their pre-images.

arXiv comment28 pages, 7 figures