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Measure-preservation criteria for a certain class of 1-lipschitz functions on Zp in mahler's expansion
Conjugacies of model sets
| 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 |
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.
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [11] |
H. Boulmezaoud, Cohomologie des pavages et leurs déformations, PhD-thesis, Lyon, 2009. |
| [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. |
| [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. |
| [14] |
J. Kellendonk,
Pattern-equivariant functions and cohomology, J. Phys A., 36 (2003), 5765-5772.
doi: 10.1088/0305-4470/36/21/306. |
| [15] |
J. Kellendonk,
Pattern equivariant functions, deformations and equivalence of tiling spaces, Ergodic Theory Dynam. Systems, 28 (2008), 1153-1176.
doi: 10.1017/S014338570700065X. |
| [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. |
| [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. |
| [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. |
| [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. |
| [20] |
L. Sadun,
Pattern-equivariant cohomology with integer coefficients, Ergodic Theory and Dynamical Systems, 27 (2007), 1991-1998.
doi: 10.1017/S0143385707000259. |
| [21] |
L. Sadun,
Tiling spaces are inverse limits, J. Math. Physics, 44 (2003), 5410-5414.
doi: 10.1063/1.1613041. |
| [22] |
M. Schlottmann,
Generalized model sets and dynamical systems, Directions in Mathematical Quasicrystals, CRM Monograph Series, AMS, Providence, 13 (2000), 143-159.
|
| [23] |
B. Sing, Pisot Substitutions and Beyond, PhD thesis, Universität Bielefeld, 2007. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [11] |
H. Boulmezaoud, Cohomologie des pavages et leurs déformations, PhD-thesis, Lyon, 2009. |
| [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. |
| [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. |
| [14] |
J. Kellendonk,
Pattern-equivariant functions and cohomology, J. Phys A., 36 (2003), 5765-5772.
doi: 10.1088/0305-4470/36/21/306. |
| [15] |
J. Kellendonk,
Pattern equivariant functions, deformations and equivalence of tiling spaces, Ergodic Theory Dynam. Systems, 28 (2008), 1153-1176.
doi: 10.1017/S014338570700065X. |
| [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. |
| [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. |
| [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. |
| [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. |
| [20] |
L. Sadun,
Pattern-equivariant cohomology with integer coefficients, Ergodic Theory and Dynamical Systems, 27 (2007), 1991-1998.
doi: 10.1017/S0143385707000259. |
| [21] |
L. Sadun,
Tiling spaces are inverse limits, J. Math. Physics, 44 (2003), 5410-5414.
doi: 10.1063/1.1613041. |
| [22] |
M. Schlottmann,
Generalized model sets and dynamical systems, Directions in Mathematical Quasicrystals, CRM Monograph Series, AMS, Providence, 13 (2000), 143-159.
|
| [23] |
B. Sing, Pisot Substitutions and Beyond, PhD thesis, Universität Bielefeld, 2007. |
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