Finite subdivision rules
HTML articles powered by AMS MathViewer
- by J. W. Cannon, W. J. Floyd and W. R. Parry
- Conform. Geom. Dyn. 5 (2001), 153-196
- DOI: https://doi.org/10.1090/S1088-4173-01-00055-8
- Published electronically: December 18, 2001
- PDF | Request permission
Abstract:
We introduce and study finite subdivision rules. A finite subdivision rule $\mathcal {R}$ consists of a finite 2-dimensional CW complex $S_{\mathcal {R}}$, a subdivision $\mathcal {R}(S_{\mathcal {R}})$ of $S_{\mathcal {R}}$, and a continuous cellular map $\varphi _{\mathcal {R}}\colon \thinspace \mathcal {R}(S_{\mathcal {R}}) \to S_{\mathcal {R}}$ whose restriction to each open cell is a homeomorphism. If $\mathcal {R}$ is a finite subdivision rule, $X$ is a 2-dimensional CW complex, and $f\colon \thinspace X\to S_{\mathcal {R}}$ is a continuous cellular map whose restriction to each open cell is a homeomorphism, then we can recursively subdivide $X$ to obtain an infinite sequence of tilings. We wish to determine when this sequence of tilings is conformal in the sense of Cannon’s combinatorial Riemann mapping theorem. In this setting, it is proved that the two axioms of conformality can be replaced by a single axiom which is implied by either of them, and that it suffices to check conformality for finitely many test annuli. Theorems are given which show how to exploit symmetry, and many examples are computed.References
- Philip L. Bowers and Kenneth Stephenson, A “regular” pentagonal tiling of the plane, Conform. Geom. Dyn. 1 (1997), 58–68. MR 1479069, DOI 10.1090/S1088-4173-97-00014-3 C-NC J. W. Cannon, The theory of negatively curved spaces and groups, Ergodic theory, symbolic dynamics, and hyperbolic spaces (Trieste, 1989), Oxford Univ. Press, New York, 1991, pp. 315–369.
- James W. Cannon, The combinatorial Riemann mapping theorem, Acta Math. 173 (1994), no. 2, 155–234. MR 1301392, DOI 10.1007/BF02398434
- L. Kantorovitch, The method of successive approximations for functional equations, Acta Math. 71 (1939), 63–97. MR 95, DOI 10.1007/BF02547750
- J. W. Cannon, W. J. Floyd, and W. R. Parry, Sufficiently rich families of planar rings, Ann. Acad. Sci. Fenn. Math. 24 (1999), no. 2, 265–304. MR 1724092
- J. W. Cannon and E. L. Swenson, Recognizing constant curvature discrete groups in dimension $3$, Trans. Amer. Math. Soc. 350 (1998), no. 2, 809–849. MR 1458317, DOI 10.1090/S0002-9947-98-02107-2
- F. M. Dekking, Recurrent sets, Adv. in Math. 44 (1982), no. 1, 78–104. MR 654549, DOI 10.1016/0001-8708(82)90066-4 GANT F. R. Gantmacher, The Theory of Matrices, Chelsea Publishing Company, New York, 1989. GARD M. Gardner, Mathematical games: In which “monster” curves force redefinition of the word “curve”, Scientific American 235 (1976), 124–133.
- Jack Giles Jr., Infinite-level replicating dissections of plane figures, J. Combin. Theory Ser. A 26 (1979), no. 3, 319–327. MR 535163, DOI 10.1016/0097-3165(79)90110-9
- Jack Giles Jr., Infinite-level replicating dissections of plane figures, J. Combin. Theory Ser. A 26 (1979), no. 3, 319–327. MR 535163, DOI 10.1016/0097-3165(79)90110-9
- Branko Grünbaum and G. C. Shephard, Tilings and patterns, W. H. Freeman and Company, New York, 1987. MR 857454
- Richard Kenyon, Self-replicating tilings, Symbolic dynamics and its applications (New Haven, CT, 1991) Contemp. Math., vol. 135, Amer. Math. Soc., Providence, RI, 1992, pp. 239–263. MR 1185093, DOI 10.1090/conm/135/1185093 Fractals B. B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman and Co., New York, 1977.
- Burt Rodin and Dennis Sullivan, The convergence of circle packings to the Riemann mapping, J. Differential Geom. 26 (1987), no. 2, 349–360. MR 906396 CP K. Stephenson, CirclePack, software, available from http://www.math.utk.edu/˜kens. Thurston W. P. Thurston, Groups, tilings and finite state automata, Summer 1989 AMS Colloquium Lectures, Geometry Center Preprint GCG 01. W J. Weeks, SnapPea: A computer program for creating and studying hyperbolic 3-manifolds, available from http://www.northnet.org/weeks.
Bibliographic 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 Polytechnic Institute & State University, Blacksburg, Virginia 24061
- MR Author ID: 67750
- Email: floyd@math.vt.edu
- W. R. Parry
- Affiliation: Department of Mathematics, Eastern Michigan University, Ypsilanti, Michigan 48197
- MR Author ID: 136390
- Email: walter.parry@emich.edu
- Received by editor(s): September 20, 1999
- Received by editor(s) in revised form: July 2, 2001
- Published electronically: December 18, 2001
- Additional Notes: This work was supported in part by NSF research grants and by The Geometry Center, University of Minnesota, an STC funded by NSF, DOE, and Minnesota Technology, Inc.
- © Copyright 2001 American Mathematical Society
- Journal: Conform. Geom. Dyn. 5 (2001), 153-196
- MSC (2000): Primary 20F65, 52C20; Secondary 05B45
- DOI: https://doi.org/10.1090/S1088-4173-01-00055-8
- MathSciNet review: 1875951