Area of Fatou sets of trigonometric functions

Schubert, Hendrik

doi:10.1090/S0002-9939-07-09015-6

Area of Fatou sets of trigonometric functions
HTML articles powered by AMS MathViewer

by Hendrik Schubert PDF

Proc. Amer. Math. Soc. 136 (2008), 1251-1259 Request permission

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

Alan F. Beardon, Iteration of rational functions, Graduate Texts in Mathematics, vol. 132, Springer-Verlag, New York, 1991. Complex analytic dynamical systems. MR 1128089, DOI 10.1007/978-1-4612-4422-6
Walter Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 2, 151–188. MR 1216719, DOI 10.1090/S0273-0979-1993-00432-4
P. Domínguez, Connectedness properties of Julia sets of transcendental entire functions, Complex Variables Theory Appl. 32 (1997), no. 3, 199–215. MR 1457686, DOI 10.1080/17476939708814991
Patricia Domínguez and Guillermo Sienra, A study of the dynamics of $\lambda \sin z$, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 12 (2002), no. 12, 2869–2883. MR 1956410, DOI 10.1142/S0218127402006199
A. È. Erëmenko and M. Yu. Lyubich, Dynamical properties of some classes of entire functions, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 4, 989–1020 (English, with English and French summaries). MR 1196102
P. Fatou, Sur l’itération des fonctions transcendantes Entières, Acta Math. 47 (1926), no. 4, 337–370 (French). MR 1555220, DOI 10.1007/BF02559517
Bogusława Karpińska, On the accessible points in the Julia sets of some entire functions, Fund. Math. 180 (2003), no. 1, 89–98. MR 2064852, DOI 10.4064/fm180-1-5
Bogusława Karpińska, Area and Hausdorff dimension of the set of accessible points of the Julia sets of $\lambda e^z$ and $\lambda \sin z$, Fund. Math. 159 (1999), no. 3, 269–287. MR 1680622, DOI 10.4064/fm_{1}999_{1}59_{3}_{1}_{2}69_{2}87
Curt McMullen, Area and Hausdorff dimension of Julia sets of entire functions, Trans. Amer. Math. Soc. 300 (1987), no. 1, 329–342. MR 871679, DOI 10.1090/S0002-9947-1987-0871679-3
John Milnor, Dynamics in one complex variable, Friedr. Vieweg & Sohn, Braunschweig, 1999. Introductory lectures. MR 1721240
G. Rottenfußer and D. Schleicher, Escaping points of the cosine family, ArXiv math.DS/0403012 (2004)
Dierk Schleicher, The dynamical fine structure of iterated cosine maps and a dimension paradox, Duke Math. J. 136 (2007), no. 2, 343–356. MR 2286634, DOI 10.1215/S0012-7094-07-13625-1
H. Schubert, Über das Maß der Fatoumenge trigonometrischer Funktionen, Diploma thesis, Kiel University, 2003
Norbert Steinmetz, Rational iteration, De Gruyter Studies in Mathematics, vol. 16, Walter de Gruyter & Co., Berlin, 1993. Complex analytic dynamical systems. MR 1224235, DOI 10.1515/9783110889314
Hans Töpfer, Über die Iteration der ganzen transzendenten Funktionen, insbesondere von $\sin z$ und $\cos z$, Math. Ann. 117 (1939), 65–84 (German). MR 1293, DOI 10.1007/BF01450008