An approximation theorem for maps between tiling spaces

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January 2011, 29(1): 323-326. doi: 10.3934/dcds.2011.29.323

An approximation theorem for maps between tiling spaces

Betseygail Rand ^1, and Lorenzo Sadun ^2,

1.	Department of Mathematics, Texas Lutheran University, Seguin, TX 78155, United States
2.	Department of Mathematics, The University of Texas at Austin, Austin, TX 78712

Received August 2009 Revised May 2010 Published September 2010

Abstract
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We show that every continuous map from one translationally finite tiling space to another can be approximated by a local map. If two local maps are homotopic, then the homotopy can be chosen so that every interpolating map is also local.

Keywords: Tilings, homotopy, local maps, pattern equivariant cohomology..

Mathematics Subject Classification: Primary: 52C23; Secondary: 37B05, 54H2.

Citation: Betseygail Rand, Lorenzo Sadun. An approximation theorem for maps between tiling spaces. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 323-326. doi: 10.3934/dcds.2011.29.323

References:

[1]	M. Barge, B. Diamond, J. Hunton and L. Sadun, Cohomology of substitution tiling spaces,, preprint, ().
[2]	J. Kellondonk, Pattern-equivariant functions and cohomology, J. Phys. A, 36 (2003), 1-8.
[3]	J. Kellendonk and I. Putnam, The Ruelle-Sullivan map for $\R^n$ actions, Math. Ann., 344 (2006), 693-711. doi: doi:10.1007/s00208-005-0728-1.
[4]	D. Lind and B. Marcus, "An Introduction to Symbolic Dynamics and Coding," Cambridge University Press, Cambridge, 1995. doi: doi:10.1017/CBO9780511626302.
[5]	K. Petersen, Factor maps between tiling dynamical systems, Forum Math., 11 (1999), 503-512. doi: doi:10.1515/form.1999.011.
[6]	N. Priebe, Towards a characterization of self-similar tilings via derived Voronoi tesselations, Geometriae Dedicata, 79 (2000), 239-265. doi: doi:10.1023/A:1005191014127.
[7]	C. Radin, The pinwheel tilings of the plane, Annals of Math., 139 (1994), 661-702. doi: doi:10.2307/2118575.
[8]	B. Rand, "Pattern-Equivariant Cohomology of Tiling Spaces With Rotations," Ph.D. thesis in Mathematics, University of Texas, 2006.
[9]	C. Radin and L. Sadun, Isomorphisms of hierarchical structures, Ergodic Theory and Dynamical Systems, 21 (2001), 1239-1248. doi: doi:10.1017/S0143385701001572.
[10]	L. Sadun, "Topology of Tiling Spaces," University Lecture Series of the American Mathematical Society, 46, 2008.

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References:

[1]	M. Barge, B. Diamond, J. Hunton and L. Sadun, Cohomology of substitution tiling spaces,, preprint, ().
[2]	J. Kellondonk, Pattern-equivariant functions and cohomology, J. Phys. A, 36 (2003), 1-8.
[3]	J. Kellendonk and I. Putnam, The Ruelle-Sullivan map for $\R^n$ actions, Math. Ann., 344 (2006), 693-711. doi: doi:10.1007/s00208-005-0728-1.
[4]	D. Lind and B. Marcus, "An Introduction to Symbolic Dynamics and Coding," Cambridge University Press, Cambridge, 1995. doi: doi:10.1017/CBO9780511626302.
[5]	K. Petersen, Factor maps between tiling dynamical systems, Forum Math., 11 (1999), 503-512. doi: doi:10.1515/form.1999.011.
[6]	N. Priebe, Towards a characterization of self-similar tilings via derived Voronoi tesselations, Geometriae Dedicata, 79 (2000), 239-265. doi: doi:10.1023/A:1005191014127.
[7]	C. Radin, The pinwheel tilings of the plane, Annals of Math., 139 (1994), 661-702. doi: doi:10.2307/2118575.
[8]	B. Rand, "Pattern-Equivariant Cohomology of Tiling Spaces With Rotations," Ph.D. thesis in Mathematics, University of Texas, 2006.
[9]	C. Radin and L. Sadun, Isomorphisms of hierarchical structures, Ergodic Theory and Dynamical Systems, 21 (2001), 1239-1248. doi: doi:10.1017/S0143385701001572.
[10]	L. Sadun, "Topology of Tiling Spaces," University Lecture Series of the American Mathematical Society, 46, 2008.

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