Hostname: page-component-7b7567c8c-wxvwp Total loading time: 0 Render date: 2026-11-14T04:53:03.763Z Has data issue: false hasContentIssue false
  • English
  • Français

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)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

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.

Information

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

References

[1]Baker, I. N., Kotus, J. and Yinian, L.. Iterates of meromorphic functions II: examples of wandering domains. J. London Math. Soc. (2) 2(42) (1990), 267–278.CrossRefGoogle Scholar
[2]Baker, I. N., Kotus, J. and Yinian, L.. Iterates of meromorphic functions III: preperiodic domains. Ergod. Th. & Dynam. Sys. 11 (1991), 603–618.CrossRefGoogle Scholar
[3]Baker, I. N., Kotus, J. and Yinian, L.. Iterates of meromorphic functions IV: critically finite functions. Results Math. 22 (1992), 651–656.CrossRefGoogle Scholar
[4]Bergweiler, W.. Iteration of meromorphic functions. Bull. Amer. Math. Soc. (N.S.) 29(2) (1993), 151–188.CrossRefGoogle Scholar
[5]Bergweiler, W.. Newton’s method and a class of meromorphic functions without wandering domains. Ergod. Th. & Dynam. Sys. 13 (1993), 231–247.CrossRefGoogle Scholar
[6]Bergweiler, W.. Singularities in Baker domains. Comput. Methods Funct. Theory 1(1) (2001), 41–49.CrossRefGoogle Scholar
[7]Bergweiler, W. and Eremenko, A. E.. On the singularities of the inverse to a meromorphic function of finite order. Rev. Mat. Iberoam. 11 (1995), 355–373.CrossRefGoogle Scholar
[8]Bergweiler, W., Haruta, M., Kriete, H., Meier, H. and Terglane, N.. On the limit functions of iterates in wandering domains. Ann. Acad. Sci. Fenn. Math. Ser. A. I. Math. 18 (1993), 369–375.Google Scholar
[9]Devaney, R. L. and Keen, L.. Dynamics of tangent. Dynamical Systems (Lecture Notes in Mathematics, 1342). Springer, Berlin, 1988, pp. 105–111.CrossRefGoogle Scholar
[10]Domínguez, P.. Dynamics of transcendental meromorphic functions. Ann. Acad. Sci. Fenn. Math. 23 (1998), 225–250.Google Scholar
[11]Prasad, M. G. P. and Nayak, T.. Dynamics of λtanh (e z). Discrete Contin. Dyn. Syst. 19(1) (2007), 121–138.Google Scholar
[12]Herring, M. E.. Mapping properties of Fatou components. Ann. Acad. Sci. Fenn. Math. 23 (1998), 263–274.Google Scholar
[13]Jiang, W. H.. Dynamics of λtan z. PhD Thesis, Graduate Center of the City University of New York, CUNY, 1991.Google Scholar
[14]Keen, L. and Kotus, J.. Dynamics of the family λtan z. Conform. Geom. Dyn. 1 (1997), 28–57.CrossRefGoogle Scholar
[15]König, H.. Conformal conjugacies in Baker domains. J. London Math. Soc. (2) 59(2) (1999), 153–170.CrossRefGoogle Scholar
[16]Morosowa, S.. An example of cyclic Baker domains. Mem. Fac. Sci. Kochi. Univ. Ser. A Math. 20 (1999), 123–126.Google Scholar
[17]Nevanlinna, R.. Analytic Functions. Springer, Berlin, 1970.CrossRefGoogle Scholar
[18]Rippon, P. J. and Stallard, G. M.. Singularities of meromorphic functions with Baker domains. Math. Proc. Cambridge Philos. Soc. 141 (2006), 371–382.CrossRefGoogle Scholar
[19]Rippon, P. J. and Stallard, G. M.. Iteration of a class of hyperbolic meromorphic functions. Proc. Amer. Math. Soc. 127(11) (1999), 3251–3258.CrossRefGoogle Scholar
[20]Stallard, G. M.. A class of meromorphic functions with no wandering domains. Ann. Acad. Sci. Fenn. Ser. Math. 16 (1991), 211–226.CrossRefGoogle Scholar
[21]Zheng, J. H.. On transcendental meromorphic functions which are geometrically finite. J. Aust. Math. Soc. 72 (2002), 93–107.CrossRefGoogle Scholar
[22]Zheng, J. H.. Singularities and limit functions in iteration of meromorphic functions. J. London Math. Soc. (2) 67(1) (2003), 195–207.CrossRefGoogle Scholar
[23]Zheng, J. H.. Singularities and wandering domains in iteration of meromorphic functions. Illinois J. Math. 44(3) (2000), 520–530.CrossRefGoogle Scholar