Conformal Geometry and Dynamics

Author(s): Jeremy T. Tyson
Journal: Conform. Geom. Dyn. 12 (2008), 32-57.
MSC (2000): Primary 30C65; Secondary 28A78, 43A80
Posted: March 6, 2008
Retrieve article in: PDF DVI PostScript

Abstract: We study global conformal Assouad dimension in the Heisenberg group $\mathbb{H}^n$ . For each $\alpha\in\{0\}\cup[1,2n+2]$ , there is a bounded set in $\mathbb{H}^n$ with Assouad dimension $\alpha$ whose Assouad dimension cannot be lowered by any quasiconformal map of $\mathbb{H}^n$ . On the other hand, for any set

in $\mathbb{H}^n$ with Assouad dimension strictly less than one, the infimum of the Assouad dimensions of sets

, taken over all quasiconformal maps

of $\mathbb{H}^n$ , equals zero. We also consider dilatation-dependent bounds for quasiconformal distortion of Assouad dimension. The proofs use recent advances in self-similar fractal geometry and tilings in $\mathbb{H}^n$ and regularity of the Carnot-Carathéodory distance from smooth hypersurfaces.

1.: ARCOZZI, N., AND FERRARI, F.,
Metric normal and distance function in the Heisenberg group.
Math. Z. 256, 3 (2007), 661-684. MR 2299576
2.: ARCOZZI, N., FERRARI, F., AND MONTEFALCONE, F.,
in preparation.
3.: ASSOUAD, P.,
Plongements lipschitziens dans $\mathbb{R}^n$ .
Bull. Soc. Math. France 111 (1983), 429-448. MR 763553 (86f:54050)
4.: BALOGH, Z. M.,
Hausdorff dimension distribution of quasiconformal mappings on the Heisenberg group.
J. Anal. Math. 83 (2001), 289-312. MR 1828495 (2002b:30022)
5.: BALOGH, Z. M., HOEFER-ISENEGGER, R., AND TYSON, J. T.,
Lifts of Lipschitz maps and horizontal fractals in the Heisenberg group.
Ergodic Theory Dynam. Systems 26, 3 (2006). MR 2237461 (2007i:28009)
6.: BALOGH, Z. M., AND TYSON, J. T.,
Hausdorff dimensions of self-similar and self-affine fractals in the Heisenberg group.
Proc. London Math. Soc. (3) 91, 1 (2005), 153-183. MR 2149533 (2006b:28008)
7.: BALOGH, Z. M., TYSON, J. T., AND WARHURST, B.,
Gromov's dimension comparison problem on Carnot groups.
C. R. Math. Acad. Sci. Paris 346,
(2008), 135-138.
8.: BALOGH, Z. M., TYSON, J. T., AND WARHURST, B.,
Sub-Riemannian vs. Euclidean dimension comparison and fractal geometry on Carnot groups.
Preprint, 2007.
9.: BISHOP, C. J.,
Quasiconformal mappings which increase dimension.
Ann. Acad. Sci. Fenn. Ser. A I Math. 24 (1999), 397-407. MR 1724076 (2000i:30044)
10.: BISHOP, C. J., AND TYSON, J. T.,
Conformal dimension of the antenna set.
Proc. Amer. Math. Soc. 129 (2001), 3631-3636. MR 1860497 (2002j:30021)
11.: BISHOP, C. J., AND TYSON, J. T.,
Locally minimal sets for conformal dimension.
Ann. Acad. Sci. Fenn. Ser. A I Math. 26 (2001), 361-373. MR 1833245 (2002c:30027)
12.: BONK, M., AND KLEINER, B.,
Conformal dimension and Gromov hyperbolic groups with 2-sphere boundary.
Geom. Topol. 9 (2005), 219-246 (electronic). MR 2116315 (2005k:20102)
13.: BOURDON, M.,
Au bord de certains polyèdres hyperboliques.
Ann. Inst. Fourier (Grenoble) 45 (1995), 119-141. MR 1324127 (96b:20045)
14.: BOURDON, M.,
Immeubles hyperboliques, dimension conforme et rigidité de Mostow.
Geom. Funct. Anal. 7 (1997), 245-268. MR 1445387 (98c:20056)
15.: CAPOGNA, L.,
Regularity of quasilinear equations in the Heisenberg group.
Comm. Pure Appl. Math. 50, 9 (1997), 867-889. MR 1459590 (98k:22037)
16.: CAPOGNA, L.,
Regularity for quasilinear equations and -quasiconformal maps in Carnot groups.
Math. Ann. 313, 2 (1999), 263-295. MR 1679786 (2000a:35027)
17.: CAPOGNA, L., AND COWLING, M.,
Conformality and -harmonicity in Carnot groups.
Duke Math. J. 135, 3 (2006), 455-479. MR 2272973 (2007h:30017)
18.: CAPOGNA, L., DANIELLI, D., PAULS, S. D., AND TYSON, J. T.,
An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem, vol. 259 of Progress in Mathematics.
Birkhäuser Verlag, Basel, 2007. MR 2312336
19.: CAPOGNA, L., AND GAROFALO, N.,
Ahlfors type estimates for perimeter measures in Carnot-Carathéodory spaces.
J. Geom. Anal. 16, 3 (2006), 455-497. MR 2250055 (2008a:28005)
20.: FALCONER, K. J., AND MAULDIN, R. D.,
Fubini-type theorems for general measure constructions.
Mathematika 47 (2000), 251-265. MR 1924502 (2003e:28009)
21.: GEHRING, F. W., AND VäISäLä, J.,
Hausdorff dimension and quasiconformal mappings.
J. London Math. Soc. (2) 6 (1973), 504-512. MR 0324028 (48:2380)
22.: GELBRICH, G.,
Self-similar periodic tilings on the Heisenberg group.
J. Lie Theory 4, 1 (1994), 31-37. MR 1326950 (96e:22013)
23.: HEINONEN, J.,
Calculus on Carnot groups.
In Fall School in Analysis (Jyväskylä, 1994), vol. 68. Ber. Univ. Jyväskylä Math. Inst., Jyväskylä, 1995, pp. 1-31. MR 1351042 (96j:22015)
24.: HEINONEN, J.,
A capacity estimate on Carnot groups.
Bull. Sci. Math. 119 (1995), 475-484. MR 1354248 (96j:22011)
25.: HEINONEN, J.,
Lectures on analysis on metric spaces.
Springer-Verlag, New York, 2001. MR 1800917 (2002c:30028)
26.: HEINONEN, J., AND KOSKELA, P.,
Quasiconformal maps in metric spaces with controlled geometry.
Acta Math. 181 (1998), 1-61. MR 1654771 (99j:30025)
27.: HEINONEN, J., AND SEMMES, S.,
Thirty-three yes or no questions about mappings, measures, and metrics.
Conform. Geom. Dyn. 1 (1997), 1-12. MR 1452413 (99h:28012)
28.: HUTCHINSON, J. E.,
Fractals and self-similarity.
Indiana Univ. Math. J. 30 (1981), 713-747. MR 625600 (82h:49026)
29.: KEITH, S., AND LAAKSO, T.,
Conformal Assouad dimension and modulus.
Geom. Funct. Anal. 14, 6 (2004), 1278-1321. MR 2135168 (2006g:30027)
30.: KIGAMI, J.,
Analysis on fractals, vol. 143 of Cambridge Tracts in Mathematics.
Cambridge University Press, Cambridge, 2001. MR 1840042 (2002c:28015)
31.: KORáNYI, A., AND REIMANN, H. M.,
Quasiconformal mappings on the Heisenberg group.
Invent. Math. 80 (1985), 309-338. MR 788413 (86m:32035)
32.: KORáNYI, A., AND REIMANN, H. M.,
Foundations for the theory of quasiconformal mappings on the Heisenberg group.
Adv. Math. 111 (1995), 1-87. MR 1317384 (96c:30021)
33.: KOVALEV, L. V.,
Conformal dimension does not assume values between zero and one.
Duke Math. J. 134, 1 (2006), 1-13. MR 2239342 (2007c:51016)
34.: LAGARIAS, J. C., AND WANG, Y.,
Self-affine tiles in ${\bf R}\sp n$ .
Adv. Math. 121, 1 (1996), 21-49. MR 1399601 (97d:52034)
35.: LUUKKAINEN, J.,
Assouad dimension: antifractal metrization, porous sets, and homogeneous measures.
J. Korean Math. Soc. 35 (1998), 23-76. MR 1608518 (99m:54023)
36.: MATTILA, P.,
Geometry of sets and measures in Euclidean spaces, vol. 44 of Cambridge Studies in Advanced Mathematics.
Cambridge University Press, Cambridge, 1995. MR 1333890 (96h:28006)
37.: MORAN, P. A. P.,
Additive functions of intervals and Hausdorff measure.
Proc. Cambridge Philos. Soc. 42 (1946), 15-23. MR 0014397 (7:278f)
38.: MOSTOW, G. D.,
Strong rigidity of locally symmetric spaces.
Princeton University Press, Princeton, N.J., 1973.
Annals of Mathematics Studies, No. 78. MR 0385004 (52:5874)
39.: PANSU, P.,
Une inégalité isopérimétrique sur le groupe de Heisenberg.
C. R. Acad. Sci. Paris Sér. I Math. 295, 2 (1982), 127-130. MR 676380 (85b:53044)
40.: PANSU, P.,
Dimension conforme et sphère à l'infini des variétés à courbure négative.
Ann. Acad. Sci. Fenn. Ser. A I Math. 14 (1989), 177-212. MR 1024425 (90k:53079)
41.: PANSU, P.,
Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un.
Ann. of Math. (2) 129 (1989), 1-60. MR 979599 (90e:53058)
42.: STRICHARTZ, R. S.,
Self-similarity on nilpotent Lie groups.
In Geometric analysis (Philadelphia, PA, 1991), vol. 140 of Contemp. Math. Amer. Math. Soc., Providence, RI, 1992, pp. 123-157. MR 1197594 (94e:43011)
43.: STRICHARTZ, R. S.,
Self-similarity in harmonic analysis.
J. Fourier Anal. Appl. 1, 1 (1994), 1-37. MR 1307067 (96c:42002)
44.: TYSON, J. T.,
Quasiconformality and quasisymmetry in metric measure spaces.
Ann. Acad. Sci. Fenn. Ser. A I Math. 23 (1998), 525-548. MR 1642158 (99i:30038)
45.: TYSON, J. T.,
Sets of minimal Hausdorff dimension for quasiconformal maps.
Proc. Amer. Math. Soc. 128, 11 (2000), 3361-3367. MR 1676353 (2001b:30033)
46.: TYSON, J. T.,
Lowering the Assouad dimension by quasisymmetric mappings.
Illinois J. Math. 45 (2001), 641-656. MR 1878624 (2002k:30040)
47.: TYSON, J. T., AND WU, J.-M.,
Quasiconformal dimensions of self-similar fractals.
Rev. Mat. Iberoamericana 22 (2006), 205-258. MR 2268118 (2008a:30024)
48.: VäISäLä, J.,
Porous sets and quasisymmetric maps.
Trans. Amer. Math. Soc. 299 (1987), 525-533. MR 869219 (88a:30049)

Retrieve articles in Conformal Geometry and Dynamics with MSC (2000): 30C65, 28A78, 43A80

Jeremy T. Tyson
Affiliation: Department of Mathematics, University of Illinois, West Green Street, Urbana, Illinois 61801
Email: tyson@math.uiuc.edu

DOI: 10.1090/S1088-4173-08-00177-X
PII: S 1088-4173(08)00177-X
Keywords: Quasiconformal map, conformal dimension, Assouad dimension, Heisenberg group, self-affine tiling.
Received by editor(s): August 27, 2007
Posted: March 6, 2008
Additional Notes: Research supported by NSF grant DMS 0555869
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-4173

Previous volume \| Table of contents \| Next volume Articles in press \| Most recent volume \| All volumes Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS