Porous sets and quasisymmetric maps
HTML articles powered by AMS MathViewer
- by Jussi Väisälä PDF
- Trans. Amer. Math. Soc. 299 (1987), 525-533 Request permission
Abstract:
A set $A$ in ${R^n}$ is called porous if there is $\alpha > 0$ such that every ball $\overline B (x,r)$ contains a point whose distance from $A$ is at least $\alpha r$. We show that porosity is preserved by quasisymmetric maps, in particular, by bilipschitz maps. Local versions are also given.References
- E. P. Dolženko, Boundary properties of arbitrary functions, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 3–14 (Russian). MR 0217297
- S. Granlund, P. Lindqvist, and O. Martio, $F$-harmonic measure in space, Ann. Acad. Sci. Fenn. Ser. A I Math. 7 (1982), no. 2, 233–247. MR 686642, DOI 10.5186/aasfm.1982.0717
- O. Martio and J. Sarvas, Injectivity theorems in plane and space, Ann. Acad. Sci. Fenn. Ser. A I Math. 4 (1979), no. 2, 383–401. MR 565886, DOI 10.5186/aasfm.1978-79.0413
- Jukka Sarvas, The Hausdorff dimension of the branch set of a quasiregular mapping, Ann. Acad. Sci. Fenn. Ser. A I Math. 1 (1975), no. 2, 297–307. MR 0396945, DOI 10.5186/aasfm.1975.0121
- 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
- T. Rado and P. V. Reichelderfer, Continuous transformations in analysis. With an introduction to algebraic topology, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band LXXV, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1955. MR 0079620 H. Federer, Geometric measure theory, Springer-Verlag, 1969.
- Jussi Väisälä, Quasisymmetric embeddings in Euclidean spaces, Trans. Amer. Math. Soc. 264 (1981), no. 1, 191–204. MR 597876, DOI 10.1090/S0002-9947-1981-0597876-7
Additional Information
- © Copyright 1987 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 299 (1987), 525-533
- MSC: Primary 30C60
- DOI: https://doi.org/10.1090/S0002-9947-1987-0869219-8
- MathSciNet review: 869219