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July  2017, 37(7): 3805-3830. doi: 10.3934/dcds.2017161

Conjugacies of model sets

Johannes Kellendonk 1, and  Lorenzo Sadun 2,

1. 

Université de Lyon, Université Claude Bernard Lyon 1, Institute Camille Jordan, CNRS UMR 5208, 69622 Villeurbanne, France
2. 

Department of Mathematics, The University of Texas at Austin, Austin, TX 78712, USA

Received  March 2015 Revised  February 2017 Published  April 2017

Fund Project: The work of the first author is partially supported by the ANR SubTile NT09 564112. The work of the second author is partially supported by NSF grant DMS-1101326

Let $M$ be a model set meeting two simple conditions: (1) the internal group $H$ is $\mathbb{R}^n$ (or a product of $\mathbb{R}^n$ and a finite group) and (2) the window $W$ is a finite union of disjoint polyhedra. Then any Delone set with finite local complexity (FLC) that is topologically conjugate to $M$ is mutually locally derivable (MLD) to a model set $M'$ that has the same internal group and window as $M$, but has a different projection from $H × \mathbb{R}^d$ to $\mathbb{R}^d$. In cohomological terms, this means that the group $H^1_{an}(M, \mathbb{R})$ of asymptotically negligible classes has dimension $n$. We also exhibit a counterexample when the second hypothesis is removed, constructing two topologically conjugate FLC Delone sets, one a model set and the other not even a Meyer set.

Keywords: Tilings, dynamical systems.
Mathematics Subject Classification: 37B50, 52C22.
Citation: Johannes Kellendonk, Lorenzo Sadun. Conjugacies of model sets. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3805-3830. doi: 10.3934/dcds.2017161
References:
[1]

J. E. Anderson and I. F. Putnam, Topological invariants for substitution tilings and their associated $C^*$-algebras, Ergodic Th. & Dynam. Syst., 18 (1998), 509-537.  doi: 10.1017/S0143385798100457.  Google Scholar

[2]

J. B. Aujogue, M. Barge, J. Kellendonk and D Lenz, Equicontinuous factors, proximality and Ellis semigroup for Delone sets, in, Mathematics of Aperiodic Order, 309 (2015), 137-194.  doi: 10.1007/978-3-0348-0903-0_5.  Google Scholar

[3]

M. Baake and U. Grimm, Aperiodic Order: Volume 1, A Mathematical Invitation (Encyclopedia of Mathematics and its Applications) Cambridge University Press 2013. doi: 10.1017/CBO9781139025256.  Google Scholar

[4]

M. Baake and D. Lenz, Deformation of Delone dynamical systems and pure point diffraction, Journal of Fourier Analysis and Applications, 11 (2005), 125-150.  doi: 10.1007/s00041-005-4021-1.  Google Scholar

[5]

M. Baake and R. V. Moody, Weighted Dirac combs with pure point diffraction, J. Reine Angew. Math. (Crelle), 573 (2004), 61-94.  doi: 10.1515/crll.2004.064.  Google Scholar

[6]

V. Baker, M. Barge and J. Kwapisz, Geometric realization and coincidence for reducible non-unimodular Pisot tiling spaces with an application to beta-shifts, Annales-Institut Fourier, 56 (2006), 2213-2248.  doi: 10.5802/aif.2238.  Google Scholar

[7]

M. Barge and J. Kellendonk, Proximality and pure point spectrum for tiling dynamical systems, Michigan Math. J., 62 (2013), 793-822.  doi: 10.1307/mmj/1387226166.  Google Scholar

[8]

M. Barge, J. Kellendonk and S. Schmieding, Maximal equicontinuous factors and cohomology for tiling spaces, Fundamenta Mathematicae, 218 (2012), 243-268.  doi: 10.4064/fm218-3-3.  Google Scholar

[9]

M. Barge, S. ŠStimac and R. F. Williams, Pure discrete spectrum in substitution tiling spaces, Disc. & Cont. Dynam. Sys. A, 33 (2013), 579-597.  doi: 10.3934/dcds.2013.33.579.  Google Scholar

[10]

G. Bernuau and M. Duneau, Fourier analysis of deformed model sets, Directions in Mathematical Quasicrystals, CRM Monograph Series, AMS, Providence, RI, (2000), 43-60.   Google Scholar

[11]

H. Boulmezaoud, Cohomologie des pavages et leurs déformations, PhD-thesis, Lyon, 2009. Google Scholar

[12]

H. Boulmezaoud and J. Kellendonk., Comparing different versions of tiling cohomology, Topology and its Applications, 157 (2010), 2225-2239.  doi: 10.1016/j.topol.2010.06.004.  Google Scholar

[13]

A. Clark and L. Sadun, When shape matters: Deformations of tiling spaces, Ergod. Th. & Dynam. Systems, 26 (2006), 69-86.  doi: 10.1017/S0143385705000623.  Google Scholar

[14]

J. Kellendonk, Pattern-equivariant functions and cohomology, J. Phys A., 36 (2003), 5765-5772.  doi: 10.1088/0305-4470/36/21/306.  Google Scholar

[15]

J. Kellendonk, Pattern equivariant functions, deformations and equivalence of tiling spaces, Ergodic Theory Dynam. Systems, 28 (2008), 1153-1176.  doi: 10.1017/S014338570700065X.  Google Scholar

[16]

J. Kellendonk and I. F. Putnam, The Ruelle-Sullivan map for actions of $\mathbb{R}^n$, Math. Ann., 334 (2006), 693-711.  doi: 10.1007/s00208-005-0728-1.  Google Scholar

[17]

J. Kellendonk and L. Sadun, Meyer sets, topological eigenvalues, and Cantor fiber bundles, J. London Math. Soc, 89 (2014), 114-130.  doi: 10.1112/jlms/jdt062.  Google Scholar

[18]

J.-Y. Lee, Substitution Delone sets with pure point spectrum are inter-model sets, J. Geom. Phys., 57 (2007), 2263-2285.  doi: 10.1016/j.geomphys.2007.07.003.  Google Scholar

[19]

D. Rudolph, Rectangular tilings of $\mathbb{R}^n$ and free $\mathbb{R}^n$-actions, Dynamical systems, Springer Berlin Heidelberg, 1342 (1988), 653-688.  doi: 10.1007/BFb0082854.  Google Scholar

[20]

L. Sadun, Pattern-equivariant cohomology with integer coefficients, Ergodic Theory and Dynamical Systems, 27 (2007), 1991-1998.  doi: 10.1017/S0143385707000259.  Google Scholar

[21]

L. Sadun, Tiling spaces are inverse limits, J. Math. Physics, 44 (2003), 5410-5414.  doi: 10.1063/1.1613041.  Google Scholar

[22]

M. Schlottmann, Generalized model sets and dynamical systems, Directions in Mathematical Quasicrystals, CRM Monograph Series, AMS, Providence, 13 (2000), 143-159.   Google Scholar

[23]

B. Sing, Pisot Substitutions and Beyond, PhD thesis, Universität Bielefeld, 2007. Google Scholar

show all references

References:
[1]

J. E. Anderson and I. F. Putnam, Topological invariants for substitution tilings and their associated $C^*$-algebras, Ergodic Th. & Dynam. Syst., 18 (1998), 509-537.  doi: 10.1017/S0143385798100457.  Google Scholar

[2]

J. B. Aujogue, M. Barge, J. Kellendonk and D Lenz, Equicontinuous factors, proximality and Ellis semigroup for Delone sets, in, Mathematics of Aperiodic Order, 309 (2015), 137-194.  doi: 10.1007/978-3-0348-0903-0_5.  Google Scholar

[3]

M. Baake and U. Grimm, Aperiodic Order: Volume 1, A Mathematical Invitation (Encyclopedia of Mathematics and its Applications) Cambridge University Press 2013. doi: 10.1017/CBO9781139025256.  Google Scholar

[4]

M. Baake and D. Lenz, Deformation of Delone dynamical systems and pure point diffraction, Journal of Fourier Analysis and Applications, 11 (2005), 125-150.  doi: 10.1007/s00041-005-4021-1.  Google Scholar

[5]

M. Baake and R. V. Moody, Weighted Dirac combs with pure point diffraction, J. Reine Angew. Math. (Crelle), 573 (2004), 61-94.  doi: 10.1515/crll.2004.064.  Google Scholar

[6]

V. Baker, M. Barge and J. Kwapisz, Geometric realization and coincidence for reducible non-unimodular Pisot tiling spaces with an application to beta-shifts, Annales-Institut Fourier, 56 (2006), 2213-2248.  doi: 10.5802/aif.2238.  Google Scholar

[7]

M. Barge and J. Kellendonk, Proximality and pure point spectrum for tiling dynamical systems, Michigan Math. J., 62 (2013), 793-822.  doi: 10.1307/mmj/1387226166.  Google Scholar

[8]

M. Barge, J. Kellendonk and S. Schmieding, Maximal equicontinuous factors and cohomology for tiling spaces, Fundamenta Mathematicae, 218 (2012), 243-268.  doi: 10.4064/fm218-3-3.  Google Scholar

[9]

M. Barge, S. ŠStimac and R. F. Williams, Pure discrete spectrum in substitution tiling spaces, Disc. & Cont. Dynam. Sys. A, 33 (2013), 579-597.  doi: 10.3934/dcds.2013.33.579.  Google Scholar

[10]

G. Bernuau and M. Duneau, Fourier analysis of deformed model sets, Directions in Mathematical Quasicrystals, CRM Monograph Series, AMS, Providence, RI, (2000), 43-60.   Google Scholar

[11]

H. Boulmezaoud, Cohomologie des pavages et leurs déformations, PhD-thesis, Lyon, 2009. Google Scholar

[12]

H. Boulmezaoud and J. Kellendonk., Comparing different versions of tiling cohomology, Topology and its Applications, 157 (2010), 2225-2239.  doi: 10.1016/j.topol.2010.06.004.  Google Scholar

[13]

A. Clark and L. Sadun, When shape matters: Deformations of tiling spaces, Ergod. Th. & Dynam. Systems, 26 (2006), 69-86.  doi: 10.1017/S0143385705000623.  Google Scholar

[14]

J. Kellendonk, Pattern-equivariant functions and cohomology, J. Phys A., 36 (2003), 5765-5772.  doi: 10.1088/0305-4470/36/21/306.  Google Scholar

[15]

J. Kellendonk, Pattern equivariant functions, deformations and equivalence of tiling spaces, Ergodic Theory Dynam. Systems, 28 (2008), 1153-1176.  doi: 10.1017/S014338570700065X.  Google Scholar

[16]

J. Kellendonk and I. F. Putnam, The Ruelle-Sullivan map for actions of $\mathbb{R}^n$, Math. Ann., 334 (2006), 693-711.  doi: 10.1007/s00208-005-0728-1.  Google Scholar

[17]

J. Kellendonk and L. Sadun, Meyer sets, topological eigenvalues, and Cantor fiber bundles, J. London Math. Soc, 89 (2014), 114-130.  doi: 10.1112/jlms/jdt062.  Google Scholar

[18]

J.-Y. Lee, Substitution Delone sets with pure point spectrum are inter-model sets, J. Geom. Phys., 57 (2007), 2263-2285.  doi: 10.1016/j.geomphys.2007.07.003.  Google Scholar

[19]

D. Rudolph, Rectangular tilings of $\mathbb{R}^n$ and free $\mathbb{R}^n$-actions, Dynamical systems, Springer Berlin Heidelberg, 1342 (1988), 653-688.  doi: 10.1007/BFb0082854.  Google Scholar

[20]

L. Sadun, Pattern-equivariant cohomology with integer coefficients, Ergodic Theory and Dynamical Systems, 27 (2007), 1991-1998.  doi: 10.1017/S0143385707000259.  Google Scholar

[21]

L. Sadun, Tiling spaces are inverse limits, J. Math. Physics, 44 (2003), 5410-5414.  doi: 10.1063/1.1613041.  Google Scholar

[22]

M. Schlottmann, Generalized model sets and dynamical systems, Directions in Mathematical Quasicrystals, CRM Monograph Series, AMS, Providence, 13 (2000), 143-159.   Google Scholar

[23]

B. Sing, Pisot Substitutions and Beyond, PhD thesis, Universität Bielefeld, 2007. Google Scholar

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