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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

 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

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.
Citation: Betseygail Rand, Lorenzo Sadun. An approximation theorem for maps between tiling spaces. Discrete & Continuous Dynamical Systems - A, 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. [3] J. Kellendonk and I. Putnam, The Ruelle-Sullivan map for $\R^n$ actions,, Math. Ann., 344 (2006), 693. doi: doi:10.1007/s00208-005-0728-1. [4] D. Lind and B. Marcus, "An Introduction to Symbolic Dynamics and Coding,", Cambridge University Press, (1995). doi: doi:10.1017/CBO9780511626302. [5] K. Petersen, Factor maps between tiling dynamical systems,, Forum Math., 11 (1999), 503. 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. doi: doi:10.1023/A:1005191014127. [7] C. Radin, The pinwheel tilings of the plane,, Annals of Math., 139 (1994), 661. doi: doi:10.2307/2118575. [8] B. Rand, "Pattern-Equivariant Cohomology of Tiling Spaces With Rotations,", Ph.D. thesis in Mathematics, (2006). [9] C. Radin and L. Sadun, Isomorphisms of hierarchical structures,, Ergodic Theory and Dynamical Systems, 21 (2001), 1239. doi: doi:10.1017/S0143385701001572. [10] L. Sadun, "Topology of Tiling Spaces,", University Lecture Series of the American Mathematical Society, 46 (2008).

show all references

##### 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. [3] J. Kellendonk and I. Putnam, The Ruelle-Sullivan map for $\R^n$ actions,, Math. Ann., 344 (2006), 693. doi: doi:10.1007/s00208-005-0728-1. [4] D. Lind and B. Marcus, "An Introduction to Symbolic Dynamics and Coding,", Cambridge University Press, (1995). doi: doi:10.1017/CBO9780511626302. [5] K. Petersen, Factor maps between tiling dynamical systems,, Forum Math., 11 (1999), 503. 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. doi: doi:10.1023/A:1005191014127. [7] C. Radin, The pinwheel tilings of the plane,, Annals of Math., 139 (1994), 661. doi: doi:10.2307/2118575. [8] B. Rand, "Pattern-Equivariant Cohomology of Tiling Spaces With Rotations,", Ph.D. thesis in Mathematics, (2006). [9] C. Radin and L. Sadun, Isomorphisms of hierarchical structures,, Ergodic Theory and Dynamical Systems, 21 (2001), 1239. 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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