Iteration of Certain Meromorphic Functions with Unbounded Singular Values

Abstract

Let MD f f .z/ D .zm=sinhm z/ for z 2 C j either m or m=2 is an odd natural numberg. For each f 2M, 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 D f f.z/ D f .z/ j 2 R n f0gg is investigated for each f 2M. Itis shown that, for each f 2M, there is a critical parameter > 0 depending on f suchthat a period-doubling bifurcation occurs in the dynamics of functions f in S when theparameter jj passes through . The non-existence of Baker domains and wanderingdomains in the Fatou set of f is proved. Further, it is shown that the Fatou set of f isinfinitely connected for 0 < jj whereas for jj , the Fatou set of f consists ofinfinitely many components and each component is simply connected.