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Iteration of certain meromorphic functions with unbounded singular values

Published online by Cambridge University Press:  21 July 2009

TARAKANTA NAYAK  and
M. GURU PREM PRASAD
TARAKANTA NAYAK
Affiliation:
Department of Mathematics, National Institute of Technology Rourkela, Rourkela 769008, India (email: tarakanta.nayak@nitrkl.ac.in)
M. GURU PREM PRASAD
Affiliation:
Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati 781039, India (email: mgpp@iitg.ac.in)
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Abstract

Let ℳ={f(z)=(zm/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-parameter family 𝒮={fλ(z)=λf(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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 3 , June 2010 , pp. 877 - 891
Copyright
Copyright © Cambridge University Press 2009

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