Bilipschitz homogeneous Jordan curves

Ghamsari, Manouchehr; Herron, David

doi:10.1090/S0002-9947-99-02324-7

Bilipschitz homogeneous Jordan curves
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by Manouchehr Ghamsari and David A. Herron

Trans. Amer. Math. Soc. 351 (1999), 3197-3216

DOI: https://doi.org/10.1090/S0002-9947-99-02324-7

Published electronically: March 29, 1999

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Abstract:

We characterize bilipschitz homogeneous Jordan curves by utilizing quasihomogeneous parameterizations. We verify that rectifiable bilipschitz homogeneous Jordan curves satisfy a chordarc condition. We exhibit numerous examples including a bilipschitz homogeneous quasicircle which has lower Hausdorff density zero. We examine homeomorphisms between Jordan curves.

References

V. V. Aseev and A. A. Shalaginov, Mappings that boundedly distort distance ratios, Dokl. Akad. Nauk 335 (1994), no. 2, 133–134 (Russian); English transl., Russian Acad. Sci. Dokl. Math. 49 (1994), no. 2, 248–250. MR 1288830
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
Timo Erkama, On domains of bounded dilatation, Complex analysis—fifth Romanian-Finnish seminar, Part 1 (Bucharest, 1981) Lecture Notes in Math., vol. 1013, Springer, Berlin, 1983, pp. 68–75. MR 738082, DOI 10.1007/BFb0066518
K. J. Falconer, The geometry of fractal sets, Cambridge Tracts in Mathematics, vol. 85, Cambridge University Press, Cambridge, 1986. MR 867284
Kenneth Falconer, Fractal geometry, John Wiley & Sons, Ltd., Chichester, 1990. Mathematical foundations and applications. MR 1102677
F. W. Gehring and B. P. Palka, Quasiconformally homogeneous domains, J. Analyse Math. 30 (1976), 172–199. MR 437753, DOI 10.1007/BF02786713
Manouchehr Ghamsari, Quasiconformal groups acting on $B^3$ that are not quasiconformally conjugate to Möbius groups, Ann. Acad. Sci. Fenn. Ser. A I Math. 20 (1995), no. 2, 245–250. MR 1346809
M.Ghamsari and D.A.Herron, Higher dimensional Ahlfors regular sets and chordarc curves in $\mathbb {R}^n$, Rocky Mountain J. Math., to appear.
John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713–747. MR 625600, DOI 10.1512/iumj.1981.30.30055
Olli Lehto, Quasiconformal mappings in the plane, Lectures on quasiconformal mappings, Dept. Math., Univ. Maryland, Lecture Note, No. 14, Dept. Math., Univ. Maryland, College Park, Md., 1975, pp. 1–43. MR 0393472
P.MacManus, R.Näkki and B.P.Palka, Quasiconformally homogeneous compacta in the complex plane, Michigan Math. J. 45 (1998) 227–241.
Pertti Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. MR 1333890, DOI 10.1017/CBO9780511623813
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
Jukka Sarvas, Boundary of a homogeneous Jordan domain, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 511–514. MR 802515, DOI 10.5186/aasfm.1985.1057
A. A. Shalaginov, Mappings of self-similar curves, Sibirsk. Mat. Zh. 34 (1993), no. 6, 210–215, v, x (Russian, with English and Russian summaries); English transl., Siberian Math. J. 34 (1993), no. 6, 1190–1195. MR 1268173, DOI 10.1007/BF00973484
Pekka Tukia, A quasiconformal group not isomorphic to a Möbius group, Ann. Acad. Sci. Fenn. Ser. A I Math. 6 (1981), no. 1, 149–160. MR 639972, DOI 10.5186/aasfm.1981.0625
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
Pekka Tukia and Jussi Väisälä, Bi-Lipschitz extensions of maps having quasiconformal extensions, Math. Ann. 269 (1984), no. 4, 561–572. MR 766014, DOI 10.1007/BF01450765
Jussi Väisälä, Quasiconformal maps of cylindrical domains, Acta Math. 162 (1989), no. 3-4, 201–225. MR 989396, DOI 10.1007/BF02392837