J. Aust. Math. Soc. 83 (2007), no. 3, pp. 369–384.

Permutable functions concerning differential equations

X. Hua R. Vaillancourt X. L. Wang
Department of Mathematics and Statistics
University of Ottawa
Ottawa, ON, K1N 6N5
Canada
hua@mathstat.uottawa.ca
Department of Mathematics and Statistics
University of Ottawa
Ottawa, ON, K1N 6N5
Canada
remi@ottawa.ca
Department of Applied Mathematics
Nanjing University of Finance and Economics
Nanjing 210003, Jiangsu
China
wangxiaoling@vip.163.com
Received 25 January 2006; revised 28 June 2006
Communicated by P. Fenton
This research was partially supported by the Natural Sciences and Engineering Research Council of Canada
and NNSF of China, No: 10371069.

Abstract

Let f and g be two permutable transcendental entire functions. Assume that f is a solution of a linear differential equation with polynomial coefficients. We prove that, under some restrictions on the coefficients and the growth of f and g, there exist two non-constant rational functions R_1 and R_2 such that R_1(f)=R_2(g). As a corollary, we show that f and g have the same Julia set: J(f)=J(g). As an application, we study a function f which is a combination of exponential functions with polynomial coefficients. This research addresses an open question due to Baker.

Download the article in PDF format (size 148 Kb)

2000 Mathematics Subject Classification: primary 30D05, 37F10, 37F50; secondary 34A20
(Metadata: XML, RSS, BibTeX)
indicates author for correspondence

References

  1. I. N. Baker, ‘Wandering domains in the iteration of entire functions’, Proc. London Math. Soc. 3 (1984), 563–576. MR759304
  2. W. Bergweiler and A. Hinkkanen, ‘On semiconjugation of entire functions’, Math. Proc. Cambridge Philos. Soc. 126 (1999), 565–574. MR1684251
  3. C. T. Chuang and C. C. Yang, Fix-Points and Factorization of Meromorphic Functions (World Scientific, Singapore, 1990). MR1050548
  4. P. Fatou, ‘Sur les équations fonctionelles’, Bull. Soc. Math. France 47 (1919), 161–271. MR1504787
  5. P. Fatou, ‘Sur les équations fonctionelles’, Bull. Soc. Math. France 48 (1920), 33–94. MR1504792
  6. P. Fatou, ‘Sur les équations fonctionelles’, Bull. Soc. Math. France 48 (1920), 208–314. MR1504797
  7. F. Gross and C. F. Osgood, ‘On fixed points of composite entire functions’, J. London Math. Soc. 28 (1983), 57–61. MR703464
  8. W. K. Hayman, Meromorphic functions (Clarendon Press, Oxford, 1964). MR164038
  9. X. H. Hua and C. C. Yang, Dynamics of transcendental functions (Gordon and Breach Science Publishers, 1998). MR1652248
  10. G. Julia, ‘Mémoire sur la permutabilité des fractions rationnelles’, Ann. Sci. École Norm. Sup. 39 (1922), 131–215. MR1509242
  11. I. Laine, Nevanlinna Theory and Complex Differential Equations (Walter de Gruyter, Berlin-New York, 1993). MR1207139
  12. L. W. Liao and C. C. Yang, ‘Some further results on the Julia sets of two permutable entire functions’, Rocky Mountain J. Math. (to appear). MR2206028
  13. S. Morosawa, Y. Nishimura, M. Taniguchi and T. Ueda, Holomorphic Dynamics (Cambridge University Press, 2000). MR1747010
  14. T. W. Ng, ‘Permutable entire functions and their Julia sets’, Math. Proc. Cambridge Philos. Soc. 131 (2001), 129–138. MR1833078
  15. G. Polya and G. Szegö, Problems and Theorems in Analysis I (Springer, New York, 1972). MR1492447
  16. K. K. Poon and C. C. Yang, ‘Dynamical behavior of two permutable entire functions’, Ann. Polon. Math. 168 (1998), 159–163. MR1610556
  17. X. L. Wang, X. H. Hua, C. C. Yang and D. G. Yang, ‘Dynamics of permutable transcendental entire functions’, Rocky Mountain J. Math. 36 (2006), 2041–2055. MR2305645
  18. X. L. Wang and C. C. Yang, ‘On the Fatou components of two permutable transcendental entire functions’, J. Math. Anal. Appl. 278 (2003), 512–526. MR1974022
  19. J. H. Zheng and Z. Z. Zhou, ‘Permutability of entire functions satisfying certain differential equations’, Tohoku Math. J. 40 (1988), 323–330. MR957047
Australian Mathematical Publishing Association Inc.

Valid XHTML 1.0 Transitional

Feedback