The Journal of the Indian Mathematical Society

On Dynamics of Semiconjugated Entire Functions


Affiliations

  • University of Delhi, Department of Mathematics, Delhi, 110007, India

Abstract

Let g and h be transcendental entire functions and let f be a continuous map of the complex plane into itself with f◦g = h◦f. Then g and h are said to be semiconjugated by f and f is called a semiconjugacy. We consider the dynamical properties of semiconjugated transcendental entire functions g and h and provide several conditions under which the semiconjugacy f carries Fatou set of one entire function into the Fatou set of other entire function appearing in the semiconjugation. We have also shown that under certain condition on the growth of entire functions appearing in the semiconjugation, the set of asymptotic values of the derivative of composition of the entire functions is bounded.

Keywords

Semiconjugation, Normal Family, Wandering Domain, Bounded Type, Permutable, Asymptotic Value.

Subject Discipline

Mathematical Sciences

Full Text:

References

I. N. Baker, Wandering domains in the iteration of entire functions, Proc. London Math. Soc. 49 (1984), 563–576.

I. N. Baker, J. Kotus and Lu. Yinian, Iterates of meromorphic functions I, Ergodic Theory and Dynamical Systems, 11 (1991), 241–248.

I. N. Baker, J. Kotus and Lu Yinian, Iterates of meromorphic functions IV: Critically finite functions, Results Math. 22 (1992), 651–656.

A. F. Beardon, Iteration of rational functions, Springer Verlag, Berlin 1991.

W. Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc. 29 (1993), 151–188.

W. Bergweiler, M. Haruta, H. Kriete, H. G. Meier and N. Terglane, On the limit functions of iterates in wandering domain, Ann. Acad. Sci. Fenn. Ser. A. I. Math. 18 (1993), 369–375.

W. Bergweiler and St. Rohde, Omitted values in domains of normality, Proc. Amer. Math. Soc. 123 (1995), 1857–1858.

W. Bergweiler and A. E. Eremenko, On the singularities of the inverse to a meromorphic function of finite order, Revista Matematica Iberoamericana 11 (1995), 355–373.

W. Bergweiler and Y. Wang, On the dynamics of composite entire functions, Ark. Math. 36 (1998), 31–39.

W. Bergweiler and A. Hinkkanen, On semiconjugation of entire functions, Math. Proc. Camb. Phil. Soc. 129 (1999), 565–574.

C. J. Bishop, Constructing entire functions by quasiconformal folding, preprint (2011).

A. E. Eremenko and M. Yu. Lyubich, Dynamical properties of some classes of entire functions, Ann. Inst. Fourier, Grenoble, 42 (1992), 989–1020.

L. R. Goldberg and L. Keen, A finiteness theorem for a dynamical class of entire functions, Ergodic Theory and Dynamical Systems, 6 (1986), 183–192.

M. E. Herring, Mapping properties of Fatou components, Ann. Acad. Sci. Fenn. Math. 23 (1998), 263–274.

X. H. Hua and C. C. Yang, Dynamics of transcendental functions, Gordon and Breach Science Pub. 1998.

D. Kumar, G. Datt and S. Kumar, Dynamics of composite entire functions, arXiv:math.DS/12075930, (2013), submitted for publication.

D. Kumar and S. Kumar, On dynamics of composite entire functions and singularities, arXiv:math.DS/13075785, (2013), submitted for publication.

K. K. Poon and C. C. Yang, Dynamics of composite entire functions, Proc. Japan. Acad. Sci. 74 (1998), 87–89.

R. Sharma and A. K. Sharma, On the growth of semiconjugated entire functions, International Mathematical Forum 3 (2008), 2175–2179.

A. P. Singh and K. Singh, On semiconjugation of entire functions, Bull. Pure Appl. Math. 2 (2008), 25–29.

G. W. Stallard, A class of meromorphic functions with no wandering domains, Ann. Acad. Sci. Fenn. Ser. A. I. Math. 16 (1991), 211–226.

D. Sullivan, Quasiconformal homeomorphism and dynamics I, Solution of the Fatou Julia problem on wandering domains, Ann. Math. 122 (1985), 401–418.


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