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.