Rigidity in holomorphic and quasiregular dynamics
HTML articles powered by AMS MathViewer
- by Gaven J. Martin and Volker Mayer
- Trans. Amer. Math. Soc. 355 (2003), 4349-4363
- DOI: https://doi.org/10.1090/S0002-9947-03-03160-X
- Published electronically: July 2, 2003
- PDF | Request permission
Abstract:
We consider rigidity phenomena for holomorphic functions and then more generally for uniformly quasiregular maps.References
- F. Berteloot and J.-J. Loeb, Spherical hypersurfaces and Lattès rational maps, J. Math. Pures Appl. (9) 77 (1998), no. 7, 655–666 (English, with English and French summaries). MR 1645073, DOI 10.1016/S0021-7824(98)80003-2
- Lennart Carleson, Peter W. Jones, and Jean-Christophe Yoccoz, Julia and John, Bol. Soc. Brasil. Mat. (N.S.) 25 (1994), no. 1, 1–30. MR 1274760, DOI 10.1007/BF01232933
- M. Denker, R. D. Mauldin, Z. Nitecki, and M. Urbański, Conformal measures for rational functions revisited, Fund. Math. 157 (1998), no. 2-3, 161–173. Dedicated to the memory of Wiesław Szlenk. MR 1636885, DOI 10.4064/fm_{1}998_{1}57_{2}-3_{1}_{1}61_{1}73
- A. È. Erëmenko and M. Yu. Lyubich, The dynamics of analytic transformations, Algebra i Analiz 1 (1989), no. 3, 1–70 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 3, 563–634. MR 1015124
- Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
- Peter Haïssinsky, Rigidity and expansion for rational maps, J. London Math. Soc. (2) 63 (2001), no. 1, 128–140. MR 1802762, DOI 10.1112/S0024610700001563
- Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 514561
- N. V. Ivanov, Action of Möbius transformations on homeomorphisms: stability and rigidity, Geom. Funct. Anal. 6 (1996), no. 1, 79–119. MR 1371232, DOI 10.1007/BF02246768
- Tadeusz Iwaniec and Gaven Martin, Quasiregular semigroups, Ann. Acad. Sci. Fenn. Math. 21 (1996), no. 2, 241–254. MR 1404085
- O. Lehto and K. I. Virtanen, Quasiconformal mappings in the plane, 2nd ed., Die Grundlehren der mathematischen Wissenschaften, Band 126, Springer-Verlag, New York-Heidelberg, 1973. Translated from the German by K. W. Lucas. MR 0344463, DOI 10.1007/978-3-642-65513-5
- M. Yu. Lyubich, Dynamics of rational transformations: topological picture, Uspekhi Mat. Nauk 41 (1986), no. 4(250), 35–95, 239 (Russian). MR 863874
- Mikhail Lyubich and Yair Minsky, Laminations in holomorphic dynamics, J. Differential Geom. 47 (1997), no. 1, 17–94. MR 1601430
- R. Mañé, P. Sad, and D. Sullivan, On the dynamics of rational maps, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 2, 193–217. MR 732343, DOI 10.24033/asens.1446
- G. J. Martin, Branch sets of uniformly quasiregular maps, Conform. Geom. Dyn. 1 (1997), 24–27. MR 1454921, DOI 10.1090/S1088-4173-97-00016-7
- G.J. Martin and V. Mayer, Local Dynamics of Uniformly Quasiregular mappings, preprint.
- O. Martio and U. Srebro, Periodic Quasimeromorphic Mappings in $\mathbb {R}^n$, J. d’Analyse Math. 28 (1975), 20-40.
- Volker Mayer, Uniformly quasiregular mappings of Lattès type, Conform. Geom. Dyn. 1 (1997), 104–111. MR 1482944, DOI 10.1090/S1088-4173-97-00013-1
- Volker Mayer, Quasiregular analogues of critically finite rational functions with parabolic orbifold, J. Anal. Math. 75 (1998), 105–119. MR 1655826, DOI 10.1007/BF02788694
- 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
- Curtis T. McMullen, Complex dynamics and renormalization, Annals of Mathematics Studies, vol. 135, Princeton University Press, Princeton, NJ, 1994. MR 1312365
- Curtis T. McMullen, Hausdorff dimension and conformal dynamics. II. Geometrically finite rational maps, Comment. Math. Helv. 75 (2000), no. 4, 535–593. MR 1789177, DOI 10.1007/s000140050140
- Curtis T. McMullen and Dennis P. Sullivan, Quasiconformal homeomorphisms and dynamics. III. The Teichmüller space of a holomorphic dynamical system, Adv. Math. 135 (1998), no. 2, 351–395. MR 1620850, DOI 10.1006/aima.1998.1726
- Ruth Miniowitz, Normal families of quasimeromorphic mappings, Proc. Amer. Math. Soc. 84 (1982), no. 1, 35–43. MR 633273, DOI 10.1090/S0002-9939-1982-0633273-X
- Feliks Przytycki, Conical limit set and Poincaré exponent for iterations of rational functions, Trans. Amer. Math. Soc. 351 (1999), no. 5, 2081–2099. MR 1615954, DOI 10.1090/S0002-9947-99-02195-9
- Mary Rees, Positive measure sets of ergodic rational maps, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 3, 383–407. MR 870689, DOI 10.24033/asens.1511
- Seppo Rickman, Quasiregular mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 26, Springer-Verlag, Berlin, 1993. MR 1238941, DOI 10.1007/978-3-642-78201-5
- Joel L. Schiff, Normal families, Universitext, Springer-Verlag, New York, 1993. MR 1211641, DOI 10.1007/978-1-4612-0907-2
- Michael Shub and Dennis Sullivan, Expanding endomorphisms of the circle revisited, Ergodic Theory Dynam. Systems 5 (1985), no. 2, 285–289. MR 796755, DOI 10.1017/S014338570000290X
- Pekka Tukia, On quasiconformal groups, J. Analyse Math. 46 (1986), 318–346. MR 861709, DOI 10.1007/BF02796595
- Lawrence Zalcman, A heuristic principle in complex function theory, Amer. Math. Monthly 82 (1975), no. 8, 813–817. MR 379852, DOI 10.2307/2319796
Bibliographic Information
- Gaven J. Martin
- Affiliation: Department of Mathematics, University of Auckland, Auckland, New Zealand
- MR Author ID: 120465
- Email: martin@math.auckland.ac.nz
- Volker Mayer
- Affiliation: UMR 8524 du CNRS - UFR de Mathématiques, Université des Sciences et Technologies de Lille, 59655 Villeneuve d’Ascq Cedex, France
- MR Author ID: 333982
- Email: volker.mayer@univ-lille1.fr
- Received by editor(s): October 19, 1999
- Published electronically: July 2, 2003
- Additional Notes: This research was partially supported by a grant from the Marsden Fund (NZ)
- © Copyright 2003 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 355 (2003), 4349-4363
- MSC (2000): Primary 30C65; Secondary 37F45
- DOI: https://doi.org/10.1090/S0002-9947-03-03160-X
- MathSciNet review: 1990755