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Title: Iteration of Certain Meromorphic Functions with Unbounded Singular Values
Authors: Nayak, T
Prasad, M Guru Prem
Issue Date: Jul-2009
Publisher: Cambridge University Press
Citation: Ergodic Theory and Dynamical Systems, 21 July 2009, Pages 1-15
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. It is shown that, for each f 2M, there is a critical parameter  > 0 depending on f such that a period-doubling bifurcation occurs in the dynamics of functions f in S when the parameter jj passes through . The non-existence of Baker domains and wandering domains in the Fatou set of f is proved. Further, it is shown that the Fatou set of f is infinitely connected for 0 < jj   whereas for jj  , the Fatou set of f consists of infinitely many components and each component is simply connected.
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