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Twist points of planar domains
Authors:
Nicola Arcozzi, Enrico Casadio Tarabusi, Fausto Di Biase and Massimo A. Picardello
Journal:
Trans. Amer. Math. Soc. 358 (2006), 2781-2798
MSC (2000):
Primary 31A15; Secondary 30C85
Posted:
December 20, 2005
MathSciNet review:
2204056
Full-text PDF Free Access
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Additional Information
Abstract: We establish a potential theoretic approach to the study of twist points in the boundary of simply connected planar domains.
References
- 1.
N. Arcozzi, E. Casadio Tarabusi, F. Di Biase and M. Picardello, On the McMillan Twist Theorem: a potential theoretic approach, Preprint number 70, Chalmers Institute of Technology, October 2001. http://www.math.chalmers.se/Math/Research/Preprints/.
- 2.
A. Baernstein II, Ahlfors and conformal invariants, Ann. Acad. Sci. Fenn. Ser. A I Math. 13, (1988), 3, 289-312. MR 0994466 (90h:30060)
- 3.
Ch. Bishop, How geodesics approach the boundary in a simply connected domain. J. Anal. Math. 64 (1994), 291-325. MR 1303516 (96d:30028)
- 4.
K. Burdzy, My favorite open problems, http://www.math.washington.edu/~burdzy/ open.shtml.
- 5.
A. Cantón, A. Granados and Ch. Pommerenke, Conformal images of Borel sets, Bull. London Math. Soc. 35 (2003), no. 4, 479-483. MR 1979001 (2004d:28002)
- 6.
J.J. Carmona and Ch. Pommerenke, Twisting behaviour of conformal maps, J. London Math. Soc. (2) 56 (1997) 1, 16--36. MR 1462823 (98h:30010)
- 7.
J.J. Carmona and Ch. Pommerenke, On the argument oscillation of conformal maps, Michigan Math. J. 46 (1999), no. 2, 297-312. MR 1704213 (2000i:30014)
- 8.
J.J. Carmona and Ch. Pommerenke, On prime ends and plane continua, J. London Math. Soc. (2) 66 (2002), no. 3, 641-650. MR 1934297 (2003i:30013)
- 9.
H. Cartan, Formes différentielles. Applications élémentaires au calcul des variations et à la théorie des courbes et des surfaces, Hermann, Paris 1967. MR 0231303 (37:6858)
- 10.
M. Chuaqui and Ch. Pommerenke, An integral representation formula of the Schwarzian derivative, Comput. Methods Funct. Theory 1 (2001), no. 1, 155-163. MR 1931608 (2003j:30055)
- 11.
R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. II, Partial Differential Equations, Reprint of the 1962 original, Wiley Classics Library, John Wiley & Sons, New York, 1989. MR 1013360 (90k:35001)
- 12.
P. Davis and H. Pollak, On the zeros of total sets of polynomials, Trans. Amer. Math. Soc. 72 (1952), 82-103. MR 0045245 (13:552a)
- 13.
F. Di Biase, B. Fischer, and R. L. Urbanke, Twist points of the von Koch snowflake, Proc. Amer. Math. Soc. 126 (1998), 1487-1490. MR 1443822 (98k:31001)
- 14.
J.L. Doob, Classical potential theory and its probabilistic counterpart. Springer-Verlag, New York, 1984. MR 0731258 (85k:31001)
- 15.
P. Fatou, Séries trigonométriques et séries de Taylor, Acta Math. 30 (1906), 335-400.
- 16.
F.W. Gehring and B.G. Osgood, Uniform domains and the quasihyperbolic metric, J. d'Analyse Math. 36 (1979), 50-74. MR 0581801 (81k:30023)
- 17.
F.W. Gehring and B.P. Palka, Quasiconformally homogeneous domains, J. d'Analyse Math. 30 (1976) 172-199. MR 0437753 (55:10676)
- 18.
G.M. Goluzin, Geometric Theory of Functions of a Complex Variable, Transl. Math. Mon., Vol. 26, Amer. Math. Soc., 1969. MR 0247039 (40:308)
- 19.
M.J. Greenberg and J.R. Harper, Algebraic Topology. A first course, Benjamin, 1981. MR 643101 (83b:55001)
- 20.
H. Grunsky, Neue Abschätzungen zur konformen Abbildungen ein - und mehrfach zusammenhängender Berichte, Schr. Math. Seminars Inst. Angew. Math. Univ. Berlin 1 (1932) 95-140.
- 21.
E. Hille Analytic function theory, Vol. 1, Chelsea, 1982.
- 22.
D. S. Jerison and C. E. Kenig, Boundary behavior of harmonic functions in nontangentially accessible domains, Adv. in Math. 46 (1982), 80-147. MR 0676988 (84d:31005b)
- 23.
J. G. Llorente, Boundary values of harmonic Bloch functions in Lipschitz domains: a martingale approach, Potential Anal. 9 (1998), 229-260. MR 1666891 (2000f:31010)
- 24.
A.I. Markushevich, Theory of functions of a complex variable. Chelsea, New York, 1977. MR 0444912 (56:3258)
- 25.
J. E. McMillan, Boundary behavior of a conformal mapping, Acta Math. 123 (1969), 43-67. MR 0257330 (41:1981)
- 26.
R. Narasimhan and Y. Nievergelt, Complex analysis in one variable, Birkhäuser, Boston, 2001. MR 1803086 (2002e:30001)
- 27.
A.I. Plessner, Uber das Verhalten analytischer Funktionen am Rande ihres Definitionsbereichs, J. Reine Angew. Math. 158 (1927) 219-227.
- 28.
Ch. Pommerenke Univalent Functions, Vandenhoeck & Ruprecht, Göttingen, 1975. MR 0507768 (58:22526)
- 29.
Ch. Pommerenke, Boundary behaviour of conformal mappings, in Aspects of contemporary complex analysis (Proc. NATO Adv. Study Inst., Univ. Durham, Durham, 1979), 313-331, Academic Press, London, (1980). MR 0623475 (82i:30008)
- 30.
Ch. Pommerenke, On ergodic properties of inner functions, Math. Ann. 256 (1981), no. 1, 43-50. MR 0620121 (82h:30019)
- 31.
Ch. Pommerenke, Über das Randverhalten schlichter Funktionen, in Mathematical papers given on the occasion of Ernst Mohr's 75th birthday, 199-209, Tech. Univ. Berlin, Berlin, (1985). MR 0809004 (87f:30083)
- 32.
Ch. Pommerenke, On the boundary continuity of conformal maps, Pacific J. Math. 120, (1985), no. 2, 423-430. MR 0810781 (87c:30011)
- 33.
Ch. Pommerenke, On the boundary derivative for univalent functions, Math. Ann. 276 (1987), no. 3, 499-504. MR 0875343 (88f:30047)
- 34.
Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Grundlehren Math. Wiss., vol. 299, Springer-Verlag, Berlin, 1992. MR 1217706 (95b:30008)
- 35.
Ch. Pommerenke, and S.E. Warschawski, On the quantitative boundary behavior of conformal maps, Comment. Math. Helv. 57 (1982), no. 1, 107-129. MR 0672848 (83k:30008)
- 36.
W. Seidel, Über die Ränderzuordnung bei konformen Abbildungen, Math. Ann. 104 (1931), 182-243.
- 37.
H. von Koch, Une méthode géométrique élémentaire pour l'étude de certains questions de la théorie des courbes planes, Acta Math. 30 (1906), 145-174.
- 38.
J.L. Walsh, Note on the shape of level curves of Green's function, Amer. Math. Monthly 60 (1953), 671-674. MR 0058789 (15:424f)
- 39.
N. Wiener, Certain notions in potential theory, J. Math. Phys. Mass. Inst. Tech. 3 (1924), 24-51.
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Additional Information
Nicola Arcozzi
Affiliation:
Dipartimento di Matematica, Università di Bologna, Piazza Porta S. Donato 5, 40126 Bologna, Italy
Email:
arcozzi@dm.unibo.it
Enrico Casadio Tarabusi
Affiliation:
Dipartimento di Matematica ``G. Castelnuovo'', Università di Roma ``La Sapienza'', Piazzale A. Moro 2, 00185 Roma, Italy
Email:
casadio@mat.uniroma1.it
Fausto Di Biase
Affiliation:
Dipartimento di Scienze, Università ``G. d'Annunzio'', Viale Pindaro 87, 65127 Pescara, Italy
Email:
dibiasef@member.ams.org
Massimo A. Picardello
Affiliation:
Dipartimento di Matematica, Università di Roma ``Tor Vergata'', Via della Ricerca Scientifica, 00133 Roma, Italy
Email:
picard@mat.uniroma2.it
DOI:
http://dx.doi.org/10.1090/S0002-9947-05-03855-9
PII:
S 0002-9947(05)03855-9
Keywords:
Harmonic measure,
twist points,
McMillan Theorem
Received by editor(s):
September 9, 2004
Posted:
December 20, 2005
Additional Notes:
This research was supported in part by MIUR (Cofin.~2000). The third-named author acknowledges hospitality from the Chalmers University of Technology, through the \textit{Jubileumsfonden} from Göteborg University (1998--2002), from the University of Rome ``Tor Vergata'', through an INdAM grant (1997--1998), and from C.U.N.Y. (Nov-Dec 2002).
Article copyright:
© Copyright 2005 American Mathematical Society
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