American Institute of Mathematical Sciences

Advanced
August  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, 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

[1]

Scott Schmieding, Rodrigo Treviño. Random substitution tilings and deviation phenomena. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3869-3902. doi: 10.3934/dcds.2021020
[2]

El Houcein El Abdalaoui, Sylvain Bonnot, Ali Messaoudi, Olivier Sester. On the Fibonacci complex dynamical systems. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2449-2471. doi: 10.3934/dcds.2016.36.2449
[3]

Lianfa He, Hongwen Zheng, Yujun Zhu. Shadowing in random dynamical systems. Discrete & Continuous Dynamical Systems, 2005, 12 (2) : 355-362. doi: 10.3934/dcds.2005.12.355
[4]

Mauricio Achigar. Extensions of expansive dynamical systems. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3093-3108. doi: 10.3934/dcds.2020399
[5]

Fritz Colonius, Marco Spadini. Fundamental semigroups for dynamical systems. Discrete & Continuous Dynamical Systems, 2006, 14 (3) : 447-463. doi: 10.3934/dcds.2006.14.447
[6]

John Erik Fornæss. Sustainable dynamical systems. Discrete & Continuous Dynamical Systems, 2003, 9 (6) : 1361-1386. doi: 10.3934/dcds.2003.9.1361
[7]

Vieri Benci, C. Bonanno, Stefano Galatolo, G. Menconi, M. Virgilio. Dynamical systems and computable information. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 935-960. doi: 10.3934/dcdsb.2004.4.935
[8]

Mădălina Roxana Buneci. Morphisms of discrete dynamical systems. Discrete & Continuous Dynamical Systems, 2011, 29 (1) : 91-107. doi: 10.3934/dcds.2011.29.91
[9]

Josiney A. Souza, Tiago A. Pacifico, Hélio V. M. Tozatti. A note on parallelizable dynamical systems. Electronic Research Announcements, 2017, 24: 64-67. doi: 10.3934/era.2017.24.007
[10]

Philippe Marie, Jérôme Rousseau. Recurrence for random dynamical systems. Discrete & Continuous Dynamical Systems, 2011, 30 (1) : 1-16. doi: 10.3934/dcds.2011.30.1
[11]

Tobias Wichtrey. Harmonic limits of dynamical systems. Conference Publications, 2011, 2011 (Special) : 1432-1439. doi: 10.3934/proc.2011.2011.1432
[12]

Yejuan Wang, Chengkui Zhong, Shengfan Zhou. Pullback attractors of nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems, 2006, 16 (3) : 587-614. doi: 10.3934/dcds.2006.16.587
[13]

Alexander Sakhnovich. Dynamical canonical systems and their explicit solutions. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1679-1689. doi: 10.3934/dcds.2017069
[14]

Bernd Aulbach, Martin Rasmussen, Stefan Siegmund. Approximation of attractors of nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 215-238. doi: 10.3934/dcdsb.2005.5.215
[15]

Jérôme Rousseau, Paulo Varandas, Yun Zhao. Entropy formulas for dynamical systems with mistakes. Discrete & Continuous Dynamical Systems, 2012, 32 (12) : 4391-4407. doi: 10.3934/dcds.2012.32.4391
[16]

Giuseppe Gaeta. On the geometry of twisted prolongations, and dynamical systems. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1209-1227. doi: 10.3934/dcdss.2020070
[17]

Yujun Zhu. Preimage entropy for random dynamical systems. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 829-851. doi: 10.3934/dcds.2007.18.829
[18]

Michal Fečkan, Michal Pospíšil. Discretization of dynamical systems with first integrals. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3543-3554. doi: 10.3934/dcds.2013.33.3543
[19]

Ji Li, Kening Lu, Peter W. Bates. Invariant foliations for random dynamical systems. Discrete & Continuous Dynamical Systems, 2014, 34 (9) : 3639-3666. doi: 10.3934/dcds.2014.34.3639
[20]

Mikko Kaasalainen. Dynamical tomography of gravitationally bound systems. Inverse Problems & Imaging, 2008, 2 (4) : 527-546. doi: 10.3934/ipi.2008.2.527

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (64)
  • HTML views (50)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences