Available in electronic format
Available in print format
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826 (e) ISSN 0002-9939 (p)
     

Area of Fatou sets of trigonometric functions

Author(s): Hendrik Schubert
Journal: Proc. Amer. Math. Soc. 136 (2008), 1251-1259.
MSC (2000): Primary 37F10; Secondary 30D05
Posted: December 18, 2007
Retrieve article in: PDF DVI PostScript

Abstract | References | Similar articles | Additional information

Abstract: We extend a result of McMullen to show that the area of the Fatou set of the sine function in a vertical strip of width $ 2\pi$ is finite. This confirms a conjecture by Milnor.


References:

1.
A. F. Beardon, Iteration of rational functions, Springer, 1991 MR 1128089 (92j:30026)

2.
W. Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc. 29 (1993), 151-188 MR 1216719 (94c:30033)

3.
P. Domınguez, Connectedness properties of Julia sets of transcendental entire functions, Complex Variables Theory Appl. 32 (1997), 199-215 MR 1457686 (98f:30027)

4.
P. Domınguez and G. Sienra, A study of the dynamics of $ \lambda \sin z$, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 12 (2002), 2869-2883 MR 1956410 (2003m:37060)

5.
A. E. Eremenko and M. Yu. Lyubich, Dynamical properties of some classes of entire functions, Ann. Inst. Fourier, Grenoble 42 (1992), 989-1019 MR 1196102 (93k:30034)

6.
P. Fatou, Sur l'itération des fonctions transcendantes entières, Acta Math. 47 (1926), 337-370 MR 1555220

7.
B. Karpińska, On the accessible points in the Julia sets of some entire functions, Fund. Math. 180 (2003), 89-98 MR 2064852 (2005h:37098)

8.
B. Karpińska, Area and Hausdorff dimension of the set of accessible points in the Julia sets of $ \lambda e^{z}$ and $ \lambda \sin z$, Fund. Math. 159 (1999), 269-287 MR 1680622 (2000b:37042)

9.
C. McMullen, Area and Hausdorff dimension of Julia sets of entire functions, Trans. Amer. Math. Soc. 300 (1987), 329-342 MR 871679 (88a:30057)

10.
J. Milnor, Dynamics in one complex variable, Second Edition, Vieweg, 2000 MR 1721240 (2002i:37057)

11.
G. Rottenfußer and D. Schleicher, Escaping points of the cosine family, ArXiv math.DS/0403012 (2004)

12.
D. Schleicher, The dynamical fine structure of iterated cosine maps and a dimension paradox, Duke Math. J., 136 (2007), 343-356. MR 2286634

13.
H. Schubert, Über das Maßder Fatoumenge trigonometrischer Funktionen, Diploma thesis, Kiel University, 2003

14.
N. Steinmetz, Rational Iteration, De Gruyter, 1993 MR 1224235 (94h:30035)

15.
H. Töpfer, Über die Iteration der ganzen transzendenten Funktionen, insbesondere von $ \sin z$ und $ \cos z$, Math. Ann. 117 (1939), 65-84 MR 0001293 (1:211e)


Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 37F10, 30D05

Retrieve articles in all Journals with MSC (2000): 37F10, 30D05


Additional Information:

Hendrik Schubert
Affiliation: Department of Mathematics, Kiel University, 24098 Kiel, Germany
Email: schubert@math.uni-kiel.de

DOI: 10.1090/S0002-9939-07-09015-6
PII: S 0002-9939(07)09015-6
Received by editor(s): August 11, 2004
Received by editor(s) in revised form: November 20, 2006
Posted: December 18, 2007
Communicated by: Juha M. Heinonen
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.


  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement
Search the AMSPowered by Google