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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

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 & 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, ().   Google Scholar

[2]

J. Kellondonk, Pattern-equivariant functions and cohomology, J. Phys. A, 36 (2003), 1-8.  Google Scholar

[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.  Google Scholar

[4]

D. Lind and B. Marcus, "An Introduction to Symbolic Dynamics and Coding," Cambridge University Press, Cambridge, 1995. doi: doi:10.1017/CBO9780511626302.  Google Scholar

[5]

K. Petersen, Factor maps between tiling dynamical systems, Forum Math., 11 (1999), 503-512. doi: doi:10.1515/form.1999.011.  Google Scholar

[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.  Google Scholar

[7]

C. Radin, The pinwheel tilings of the plane, Annals of Math., 139 (1994), 661-702. doi: doi:10.2307/2118575.  Google Scholar

[8]

B. Rand, "Pattern-Equivariant Cohomology of Tiling Spaces With Rotations," Ph.D. thesis in Mathematics, University of Texas, 2006. Google Scholar

[9]

C. Radin and L. Sadun, Isomorphisms of hierarchical structures, Ergodic Theory and Dynamical Systems, 21 (2001), 1239-1248. doi: doi:10.1017/S0143385701001572.  Google Scholar

[10]

L. Sadun, "Topology of Tiling Spaces," University Lecture Series of the American Mathematical Society, 46, 2008.  Google Scholar

show all references

References:
[1]

M. Barge, B. Diamond, J. Hunton and L. Sadun, Cohomology of substitution tiling spaces,, preprint, ().   Google Scholar

[2]

J. Kellondonk, Pattern-equivariant functions and cohomology, J. Phys. A, 36 (2003), 1-8.  Google Scholar

[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.  Google Scholar

[4]

D. Lind and B. Marcus, "An Introduction to Symbolic Dynamics and Coding," Cambridge University Press, Cambridge, 1995. doi: doi:10.1017/CBO9780511626302.  Google Scholar

[5]

K. Petersen, Factor maps between tiling dynamical systems, Forum Math., 11 (1999), 503-512. doi: doi:10.1515/form.1999.011.  Google Scholar

[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.  Google Scholar

[7]

C. Radin, The pinwheel tilings of the plane, Annals of Math., 139 (1994), 661-702. doi: doi:10.2307/2118575.  Google Scholar

[8]

B. Rand, "Pattern-Equivariant Cohomology of Tiling Spaces With Rotations," Ph.D. thesis in Mathematics, University of Texas, 2006. Google Scholar

[9]

C. Radin and L. Sadun, Isomorphisms of hierarchical structures, Ergodic Theory and Dynamical Systems, 21 (2001), 1239-1248. doi: doi:10.1017/S0143385701001572.  Google Scholar

[10]

L. Sadun, "Topology of Tiling Spaces," University Lecture Series of the American Mathematical Society, 46, 2008.  Google Scholar

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