Remote Access Conformal Geometry and Dynamics
Green Open Access

Conformal Geometry and Dynamics

ISSN 1088-4173

 
 

 

Expansion complexes for finite subdivision rules. II


Authors: J. W. Cannon, W. J. Floyd and W. R. Parry
Journal: Conform. Geom. Dyn. 10 (2006), 326-354
MSC (2000): Primary 30F45, 52C20; Secondary 20F67, 52C20
DOI: https://doi.org/10.1090/S1088-4173-06-00127-5
Published electronically: December 6, 2006
MathSciNet review: 2268483
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper gives applications of earlier work of the authors on the use of expansion complexes for studying conformality of finite subdivision rules. The first application is that a one-tile rotationally invariant finite subdivision rule (with bounded valence and mesh approaching 0) has an invariant partial conformal structure, and hence is conformal. The paper next considers one-tile single valence finite subdivision rules. It is shown that an expansion map for such a finite subdivision rule can be conjugated to a linear map, and that the finite subdivision rule is conformal exactly when this linear map is either a dilation or has eigenvalues that are not real. Finally, an example is given of an irreducible finite subdivision rule that has a parabolic expansion complex and a hyperbolic expansion complex.


References [Enhancements On Off] (What's this?)

  • 1. A. F. Beardon, A Primer on Riemann Surfaces, London Mathematical Society Lecture Note Series 78, Cambridge University Press, Cambridge, 1984. MR 0808581 (87h:30090)
  • 2. E. Blanc, Propriétés génériques des laminations, Ph.D. thesis, Université Lyon-1, 2001.
  • 3. P. L. Bowers and K. Stephenson, A ``regular'' pentagonal tiling of the plane, Conform. Geom. Dyn. 1 (1997), 58-68 (electronic). MR 1479069 (99d:52016)
  • 4. J. W. Cannon, The combinatorial Riemann mapping theorem, Acta Math. 173 (1994), 155-234. MR 1301392 (95k:30046)
  • 5. J. W. Cannon, W. J. Floyd, and W. R. Parry, Squaring rectangles: the finite Riemann mapping theorem, The mathematical legacy of Wilhelm Magnus: groups, geometry and special functions (Brooklyn, NY, 1992), Amer. Math. Soc., Providence, RI, 1994, pp. 133-212. MR 1292901 (95g:20045)
  • 6. J. W. Cannon, W. J. Floyd, and W. R. Parry, Sufficiently rich families of planar rings, Ann. Acad. Sci. Fenn. 24 (1999), 265-304. MR 1724092 (2000k:20057)
  • 7. J. W. Cannon, W. J. Floyd, and W. R. Parry, Finite subdivision rules, Conform. Geom. Dyn. 5 (2001), 153-196 (electronic). MR 1875951 (2002j:52021)
  • 8. J. W. Cannon, W. J. Floyd, R. Kenyon, and W. R. Parry, Constructing rational maps from subdivision rules, Conform. Geom. Dyn. 7 (2003), 76-102 (electronic). MR 1992038 (2004f:37062)
  • 9. J. W. Cannon, W. J. Floyd, and W. R. Parry, Expansion complexes for finite subdivision rules. I, Conform. Geom. Dyn. 10 (2006), 63-99 (electronic). MR 2218641
  • 10. J. W. Cannon, W. J. Floyd, and W. R. Parry, Combinatorially regular polyomino tilings, Discrete Comp. Geom. 35 (2006), 269-285. MR 2195055 (2006h:52016)
  • 11. J. W. Cannon and E. L. Swenson, Recognizing constant curvature discrete groups in dimension 3, Trans. Amer. Math. Soc. 350 (1998), 809-849. MR 1458317 (98i:57023)
  • 12. A. Constantin and B. Kolev, The theorem of Kérékjartó on periodic homeomorphisms of the disk and the sphere, Enseign. Math. 40 (1994), 193-204. MR 1309126 (95j:55005)
  • 13. É. Ghys, Laminations par surfaces de riemann, Panoramas & Synthèses 8 (2000), 49-95. MR 1760843 (2001g:37068)
  • 14. Z.-X. He and O. Schramm, Hyperbolic and parabolic packings, Discrete. Comput. Geom. 14 (1995), 123-149. MR 1331923 (96h:52017)
  • 15. R. Kenyon, Inflationary tilings with a similarity structure, Comment. Math. Helvetici 69, (1994), 169-198. MR 1282366 (95e:52043)
  • 16. Z. Nehari, Conformal Mapping, McGraw-Hill, New York, Toronto, London, 1952. MR 0045823 (13,640h)
  • 17. K. Stephenson, CirclePack, software, available from http://www.math.utk.edu/~kens.

Similar Articles

Retrieve articles in Conformal Geometry and Dynamics of the American Mathematical Society with MSC (2000): 30F45, 52C20, 20F67, 52C20

Retrieve articles in all journals with MSC (2000): 30F45, 52C20, 20F67, 52C20


Additional Information

J. W. Cannon
Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602
Email: cannon@math.byu.edu

W. J. Floyd
Affiliation: Department of Mathematics, Virginia Tech, Blacksburg, Virginia 24061
Email: floyd@math.vt.edu

W. R. Parry
Affiliation: Department of Mathematics, Eastern Michigan University, Ypsilanti, Michigan 48197
Email: walter.parry@emich.edu

DOI: https://doi.org/10.1090/S1088-4173-06-00127-5
Keywords: Conformality, expansion complex, finite subdivision rule
Received by editor(s): November 22, 2004
Published electronically: December 6, 2006
Additional Notes: This research was supported in part by NSF grants DMS-9803868, DMS-9971783, DMS-10104030, and DMS-0203902.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society