Let M be the class of all transcendental meromorphic functions f : C → C ∪ {∞} with at least two poles or one pole that is not an omitted value, and M o = {f ∈ M : f has at least one omitted value}. Some dynamical issues of the functions in M o are addressed in this article. A complete classification in terms of forward orbits of all the multiply connected Fatou components is made. As a corollary, it follows that the Julia set is not totally disconnected unless all the omitted values are contained in a single Fatou component. Non-existence of both Baker wandering domains and invariant Herman rings are proved. Eventual connectivity of each wandering domain is proved to exist. For functions with exactly one pole, we show that Herman rings of period 2 also do not exist. A necessary and sufficient condition for the existence of a dense subset of singleton buried components in the Julia set is established for functions with two omitted values. The conjecture that a meromorphic function has at most two completely invariant Fatou components is confirmed for all f ∈ M o except in the case when f has a single omitted value, no critical value and is of infinite order. Some relevant examples are discussed.
We investigate the existence and distribution of Herman rings of transcendental meromorphic functions which have at least one omitted value. If all the poles of such a function are multiple then it has no Herman ring. Herman rings of period one or two do not exist. Functions with a single pole or with at least two poles one of which is an omitted value have no Herman ring. Every doubly connected periodic Fatou component is a Herman ring.
We define Baker omitted value, in short bov, of an entire or meromorphic function f in the complex plane as an omitted value for which there exists r 0 > 0 such that for each ball D r (a) centred at a and with radius r satisfying 0 < r < r 0 , every component of the boundary of f −1 (D r (a)) is bounded. The existence and some dynamical implications of bov is investigated in this article. The bov of a function is the only asymptotic value. An entire function has bov if and only if the image of every unbounded curve is unbounded. It follows that an entire function has bov whenever it has a Baker wandering domain. A sufficient condition for existence of bov of meromorphic functions is also proved. Functions with bov have at most one completely invariant Fatou component. Some counter examples are provided and problems are proposed for further investigation.
ARTICLE HISTORY
Abstract. Let M = { f (z) = (z m /sinh m z) for z ∈ C | either m or m/2 is an odd natural number}. For each f ∈ M, the set of singularities of the inverse function of f is an unbounded subset of the real line R. In this paper, the iteration of functions in one-It is shown that, for each f ∈ M, there is a critical parameter λ * > 0 depending on f such that a period-doubling bifurcation occurs in the dynamics of functions f λ in S when the parameter |λ| passes through λ * . The non-existence of Baker domains and wandering domains in the Fatou set of f λ is proved. Further, it is shown that the Fatou set of f λ is infinitely connected for 0 < |λ| ≤ λ * whereas for |λ| ≥ λ * , the Fatou set of f λ consists of infinitely many components and each component is simply connected.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.