Boundaries of escaping Fatou components

Rippon, P.; Stallard, G.

doi:10.1090/S0002-9939-2011-10842-6

Boundaries of escaping Fatou components
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by P. J. Rippon and G. M. Stallard PDF

Proc. Amer. Math. Soc. 139 (2011), 2807-2820 Request permission

Abstract:

Let $f$ be a transcendental entire function and $U$ be a Fatou component of $f$. We show that if $U$ is an escaping wandering domain of $f$, then most boundary points of $U$ (in the sense of harmonic measure) are also escaping. In the other direction we show that if enough boundary points of $U$ are escaping, then $U$ is an escaping Fatou component. Some applications of these results are given; for example, if $I(f)$ is the escaping set of $f$, then $I(f)\cup \{\infty \}$ is connected.

References

I. N. Baker, Completely invariant domains of entire functions, Mathematical Essays Dedicated to A. J. Macintyre, Ohio Univ. Press, Athens, Ohio, 1970, pp. 33–35. MR 0271344
I. N. Baker and P. Domínguez, Some connectedness properties of Julia sets, Complex Variables Theory Appl. 41 (2000), no. 4, 371–389. MR 1785150, DOI 10.1080/17476930008815263
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
W. Bergweiler, An entire function with simply and multiply connected wandering domains, Pure Appl. Math. Quarterly, 7 (2011), 107–120.
Walter Bergweiler and A. Hinkkanen, On semiconjugation of entire functions, Math. Proc. Cambridge Philos. Soc. 126 (1999), no. 3, 565–574. MR 1684251, DOI 10.1017/S0305004198003387
P. Domínguez, Dynamics of transcendental meromorphic functions, Ann. Acad. Sci. Fenn. Math. 23 (1998), no. 1, 225–250. MR 1601879
A. È. Erëmenko, On the iteration of entire functions, Dynamical systems and ergodic theory (Warsaw, 1986) Banach Center Publ., vol. 23, PWN, Warsaw, 1989, pp. 339–345. MR 1102727
P. Fatou, Sur l’itération des fonctions transcendantes Entières, Acta Math. 47 (1926), no. 4, 337–370 (French). MR 1555220, DOI 10.1007/BF02559517
John B. Garnett and Donald E. Marshall, Harmonic measure, New Mathematical Monographs, vol. 2, Cambridge University Press, Cambridge, 2005. MR 2150803, DOI 10.1017/CBO9780511546617
Masashi Kisaka, On the connectivity of Julia sets of transcendental entire functions, Ergodic Theory Dynam. Systems 18 (1998), no. 1, 189–205. MR 1609471, DOI 10.1017/S0143385798097570
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
Thomas Ransford, Potential theory in the complex plane, London Mathematical Society Student Texts, vol. 28, Cambridge University Press, Cambridge, 1995. MR 1334766, DOI 10.1017/CBO9780511623776
P. J. Rippon, Baker domains, Transcendental dynamics and complex analysis, London Math. Soc. Lecture Note Ser., vol. 348, Cambridge Univ. Press, Cambridge, 2008, pp. 371–395. MR 2458809, DOI 10.1017/CBO9780511735233.015
P. J. Rippon and G. M. Stallard, On questions of Fatou and Eremenko, Proc. Amer. Math. Soc. 133 (2005), no. 4, 1119–1126. MR 2117213, DOI 10.1090/S0002-9939-04-07805-0
P. J. Rippon and G. M. Stallard, Escaping points of entire functions of small growth, Math. Z. 261 (2009), no. 3, 557–570. MR 2471088, DOI 10.1007/s00209-008-0339-0
P. J. Rippon and G. M. Stallard, Slow escaping points of meromorphic functions, to appear in Trans. Amer. Math. Soc., arXiv: 0812.2410.
P. J. Rippon and G. M. Stallard, Fast escaping points of entire functions, arXiv: 1009.5081.
D. J. Sixsmith, Entire functions for which the escaping set is a spider’s web, arXiv: 1012.1303.
Walter Rudin, Real and complex analysis, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210528