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

Abstract

Iteration of the function f λ (z) = λ + z + tan z, z ∈ C is investigated in this article.It is proved that for every λ, the Fatou set of f λ has a completely invariant Baker domain B; we call it the primary Fatou component. The rest of the results deals with f λ when it is topologically hyperbolic. For all real λ or λ such that λ = πk + iλ 2 for some integer k and 0 < λ 2 < 1, the only other Fatou component is shown to be another completely invariant Baker domain.It is proved that if |2 + λ 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;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 λ for λ = kπ + i π 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.