Iteration of certain meromorphic functions with unbounded singular values

Let M= { f (z)= (zm/sinhm 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 oneparameter family S = { fλ(z)= λ f (z) | λ ∈ R \ {0}} is investigated for each f ∈M. 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 fλ in S when the parameter |λ| passes through λ. The non-existence of Baker domains and wandering domains in the Fatou set of fλ is proved. Further, it is shown that the Fatou set of fλ is infinitely connected for 0< |λ| ≤ λ whereas for |λ| ≥ λ, the Fatou set of fλ consists of infinitely many components and each component is simply connected.