Bi-Lipschitz homogeneous curves in $\mathbb {R}^2$ are quasicircles
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- by Christopher J. Bishop PDF
- Trans. Amer. Math. Soc. 353 (2001), 2655-2663 Request permission
Abstract:
We show that a bi-Lipschitz homogeneous curve in the plane must satisfy the bounded turning condition, and that this is false in higher dimensions. Combined with results of Herron and Mayer this gives several characterizations of such curves in the plane.References
- Morgan Ward, Ring homomorphisms which are also lattice homomorphisms, Amer. J. Math. 61 (1939), 783–787. MR 10, DOI 10.2307/2371336
- Beverly Brechner and Timo Erkama, On topologically and quasiconformally homogeneous continua, Ann. Acad. Sci. Fenn. Ser. A I Math. 4 (1979), no. 2, 207–208. MR 565872, DOI 10.5186/aasfm.1978-79.0402
- Timo Erkama, Quasiconformally homogeneous curves, Michigan Math. J. 24 (1977), no. 2, 157–159. MR 466539
- Manouchehr Ghamsari and David A. Herron, Higher dimensional Ahlfors regular sets and chordarc curves in $\mathbf R^n$, Rocky Mountain J. Math. 28 (1998), no. 1, 191–222. MR 1639853, DOI 10.1216/rmjm/1181071829
- Manouchehr Ghamsari and David A. Herron, Bi-Lipschitz homogeneous Jordan curves, Trans. Amer. Math. Soc. 351 (1999), no. 8, 3197–3216. MR 1608313, DOI 10.1090/S0002-9947-99-02324-7
- David A. Herron and Volker Mayer, Bi-Lipschitz group actions and homogeneous Jordan curves, Illinois J. Math. 43 (1999), no. 4, 770–792. MR 1712522
- Aarno Hohti, On Lipschitz homogeneity of the Hilbert cube, Trans. Amer. Math. Soc. 291 (1985), no. 1, 75–86. MR 797046, DOI 10.1090/S0002-9947-1985-0797046-7
- Aarno Hohti and Heikki Junnila, A note on homogeneous metrizable spaces, Houston J. Math. 13 (1987), no. 2, 231–234. MR 904953
- Paul MacManus, Raimo Näkki, and Bruce Palka, Quasiconformally bi-homogeneous compacta in the complex plane, Proc. London Math. Soc. (3) 78 (1999), no. 1, 215–240. MR 1658172, DOI 10.1112/S0024611599001690
- Paul MacManus, Raimo Näkki, and Bruce Palka, Quasiconformally homogeneous compacta in the complex plane, Michigan Math. J. 45 (1998), no. 2, 227–241. MR 1637642, DOI 10.1307/mmj/1030132180
- Volker Mayer, Trajectoires de groupes à $1$-paramètre de quasi-isométries, Rev. Mat. Iberoamericana 11 (1995), no. 1, 143–164 (French). MR 1321776, DOI 10.4171/RMI/169
- James T. Rogers Jr., Homogeneous continua, Proceedings of the 1983 topology conference (Houston, Tex., 1983), 1983, pp. 213–233. MR 738476
- James T. Rogers Jr., Classifying homogeneous continua, Proceedings of the Symposium on General Topology and Applications (Oxford, 1989), 1992, pp. 341–352. MR 1173271, DOI 10.1016/0166-8641(92)90107-B
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
- P. Tukia and J. Väisälä, Quasisymmetric embeddings of metric spaces, Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), no. 1, 97–114. MR 595180, DOI 10.5186/aasfm.1980.0531
Additional Information
- Christopher J. Bishop
- Affiliation: Department of Mathematics, SUNY at Stony Brook, Stony Brook, New York 11794-3651
- MR Author ID: 37290
- Email: bishop@math.sunysb.edu
- Received by editor(s): August 12, 1999
- Published electronically: March 14, 2001
- Additional Notes: The author is partially supported by NSF Grant DMS 98-00924
- © Copyright 2001 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 353 (2001), 2655-2663
- MSC (2000): Primary 30C65
- DOI: https://doi.org/10.1090/S0002-9947-01-02755-6
- MathSciNet review: 1828465