ON THE DISTRIBUTION OF JULIA SETS OF HOLOMORPHIC MAPS

doi:10.1007/s10473-020-0401-5

Acta mathematica scientia,Series B ›› 2020, Vol. 40 ›› Issue (4): 903-909.doi: 10.1007/s10473-020-0401-5

• Articles • Next Articles

ON THE DISTRIBUTION OF JULIA SETS OF HOLOMORPHIC MAPS

Chunlei CAO¹, Yuefei WANG^2,3

1. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China;
2. College of Mathematics and Statistics, Shenzhen University, Shenzhen 518060, China;
3. Academy of Mathemtics and Systems Sciences, Chinese Academy of Sciences, Beijing 100190, China

Received:2019-07-08 Online:2020-08-25 Published:2020-08-21
Contact: Yuefei WANG E-mail:wangyf@math.ac.cn
Supported by:
The work was supported by NSF of China (11688101).

Abstract

Abstract: In 1965 Baker first considered the distribution of Julia sets of transcendental entire maps and proved that the Julia set of an entire map cannot be contained in any finite set of straight lines. In this paper we shall consider the distribution problem of Julia sets of meromorphic maps. We shall show that the Julia set of a transcendental meromorphic map with at most finitely many poles cannot be contained in any finite set of straight lines. Meanwhile, examples show that the Julia sets of meromorphic maps with infinitely many poles may indeed be contained in straight lines. Moreover, we shall show that the Julia set of a transcendental analytic self-map of ${\bf C^*}$ can neither contain a free Jordan arc nor be contained in any finite set of straight lines.

Key words: Fatou set, Julia set, transcendental analytic map, free Jordan arc

CLC Number:

37F10

Chunlei CAO, Yuefei WANG. ON THE DISTRIBUTION OF JULIA SETS OF HOLOMORPHIC MAPS[J].Acta mathematica scientia,Series B, 2020, 40(4): 903-909.

Trendmd

References

[1] Beardon A F. Iteration of Rational Functions. New York, Berlin, Heidelberg:Springer, 1991
[2] Bergweiler W. Iteration of meromorphic functions. Bull Amer Math Soc (N S), 1993, 29:151-188
[3] McMullen C. Complex Dynamics and Renormalization. Annals of Mathematics Studies Vol 135. Princeton, NJ:Princeton University Press, 1994
[4] Fatou P. Sur l'itération des fonctions transcendantes entières. Acta Math, 1926, 47:337-370
[5] Baker I N. Sets of non-normality in iteration theory. J London Math Soc, 1965, 40:499-502
[6] Baker I N. Repulsive fixpoints of entire functions. Math Z, 1968, 104:252-256
[7] Baker I N. Completely invariant domains of entire functions//Shankar H. Mathematical Essays Dedicated to A J Macintyre. Ohio:Ohio University Press, 1970:33-35
[8] Baker I N. The domains of normality of an entire function. Ann Acad Sci Fenn (Ser A I Math), 1975, 1:277-283
[9] Baker I N. Wandering domains in the iteration of entire functions. Proc London Math Soc, 1984, 49:563-576
[10] Baker I N. Iteration of entire functions:an introductory survey//Lectures on Complex Analysis. Singapore, New Jersey, London, Hong Kong:World Scientific, 1987:1-17
[11] Rådström H. On the iteration of analytic functions. Math Scand, 1953, 1:85-92
[12] Baker I N. Wandering domains for maps of the punctured plane. Ann Acad Sci Fenn (Ser A I Math), 1987, 12:191-198
[13] Keen L. Dynamics of holomorphic self-maps of C^*//Drasin D, Earle C J, Gehring F W, Kra I, Marden A, et al. Holomorphic functions and moduli I. New York, Berlin, Heidelberg:Springer, 1988:9-30
[14] Kotus J. Iterated holomorphic maps of the punctured plane//Kurzhanski A B, Sigmund K. Dynamical systems. Lect Notes Econ & Math Syst 287. Berlin, Heidelberg, New York:Springer, 1987:10-29
[15] Baker I N, Kotus J, Lü Y. Iterates of meromorphic functions I. Ergod Th and Dynam Sys, 1991, 11:241-248
[16] Baker I N, Kotus J, Lü Y. Iterates of meromorphic functions Ⅱ:Examples of wandering domains. J London Math Soc, 1990, 42(2):267-278
[17] Baker I N, Kotus J, Lü Y. Iterates of meromorphic functions Ⅲ:Preperiodic domains. Ergod Th and Dynam Sys, 1991, 11:603-618
[18] Baker I N, Kotus J, Lü Y. Iterates of meromorphic functions IV:Critically finite functions. Results Math, 1992, 22:651-656
[19] Devaney R L, Keen L. Dynamics of meromorphic maps with polynomial Schwarzian derivative. Ann Sci École Norm Sup, 1989, 22:55-81
[20] Stallard G M. Meromorphic functions whose Julia sets contain a free Jordan arc. Ann Acad Sci Fenn (Ser A I Math), 1993, 18:273-298
[21] Qiao J Y. On the distribution of Julia sets//Yang C C, Wen G C, Li K Y, Chiang Y M, et al. Complex Analysis and Its Applications. Pitman Res Notes Math Ser 305. Harlow:Longman Sci Tech, 1994:246-257
[22] Domínguez P. Dynamics of transcendental meromorphic functions. Ann Acad Sci Fenn (Ser A I Math), 1998, 28:225-250
[23] Nevanlinna R. Analytic Functions. Berlin-Heidelberg-New York:Springer-Verlag, 1970
[24] Rippon P J, Stallard G M. Escaping points of meromorphic functions with a finite number of poles. J d'Anal Math, 2005, 96:225-245
[25] Bhattacharyya P. Iteration of Analytic Functions[D]. London:University of London, 1969

ON THE DISTRIBUTION OF JULIA SETS OF HOLOMORPHIC MAPS

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 4

Metrics

Comments

Recommended 1

[1]	Yezhou LI, Ningfang SONG. A NOTE ON q-DIFFERENCE OPERATOR AND RELATED LIMIT DIRECTION [J]. Acta mathematica scientia,Series B, 2018, 38(6): 1678-1688.
[2]	Meng YANG. HEAT KERNEL ESTIMATES ON JULIA SETS [J]. Acta mathematica scientia,Series B, 2017, 37(5): 1399-1414.
[3]	Jun WANG, Zongxuan CHEN. LIMITING DIRECTIONS OF JULIA SETS OF ENTIRE SOLUTIONS TO COMPLEX DIFFERENTIAL EQUATIONS [J]. Acta mathematica scientia,Series B, 2017, 37(1): 97-107.
[4]	Gao Junyang; Qiao Jianyong. CONTINUITY AND HAUSDORFF DIMENSION OF JULIA SET CONCERNING YANG-LEE ZEROS [J]. Acta mathematica scientia,Series B, 2008, 28(3): 530-536.