American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Boltzmann-Grad limit of a hard sphere system in a box with isotropic boundary conditions
  • DCDS Home
  • This Issue
  • Next Article
    An Erratum on "Stability and dynamics of a weak viscoelastic system with memory and nonlinear time-varying delay" (Discrete Continuous Dynamic Systems, 40(3), 2020, 1493-1515)
doi: 10.3934/dcds.2021157
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Number of bounded distance equivalence classes in hulls of repetitive Delone sets

Dirk Frettlöh 1, , Alexey Garber 2,, and  Lorenzo Sadun 3,

1. 

Technische Fakultät, Bielefeld University, Postfach 100131, 33501 Bielefeld, Germany
2. 

School of Mathematical & Statistical Sciences, The University of Texas Rio Grande Valley, 1 West University Blvd., Brownsville, TX 78520, USA
3. 

Department of Mathematics, University of Texas, 2515 Speedway, PMA 8.100, Austin, TX 78712, USA

* Corresponding author: Alexey Garber

Received  February 2021 Revised  September 2021 Early access November 2021

Fund Project: A.G. is partially supported by the Alexander von Humboldt Foundation

Two Delone sets are bounded distance equivalent to each other if there is a bijection between them such that the distance of corresponding points is uniformly bounded. Bounded distance equivalence is an equivalence relation. We show that the hull of a repetitive Delone set with finite local complexity has either one equivalence class or uncountably many.

Keywords: Delone set, finite local complexity, topological hull, bounded distance equivalence, aperiodic order, aperiodic tilings.
Mathematics Subject Classification: Primary: 37B51, 52C23.
Citation: Dirk Frettlöh, Alexey Garber, Lorenzo Sadun. Number of bounded distance equivalence classes in hulls of repetitive Delone sets. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021157
References:
[1]

J. Aliste-Prieto, D. Coronel and J. M. Gambaudo, Linearly repetitive Delone sets are rectifiable, Ann. Inst. H. Poincaré Anal. Non Lineáire, 30 (2013), 275-290.  doi: 10.1016/j.anihpc.2012.07.006.  Google Scholar

[2]

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

[3] M. Baake and U. Grimm, Aperiodic Order, Cambridge Univ. Press, Cambridge, 2013.   Google Scholar
[4]

K. Bezdek, From the Kneser–Poulsen conjecture to ball–polyhedra, European J. Combin., 29 (2008), 1820-1830.  doi: 10.1016/j.ejc.2008.01.011.  Google Scholar

[5]

B. Csikós, A Schläfli–type formula for polytopes with curved faces and its application to the Kneser–Poulsen conjecture, Monatsh. Math, 147 (2006), 273-292.  doi: 10.1007/s00605-005-0363-7.  Google Scholar

[6]

W. A. Deuber, M. Simonovits and V. T. Sós, A note on paradoxical metric spaces, Stud. Sci. Math. Hungar., 30 (1995), 17-23.   Google Scholar

[7]

M. Duneau and C. Oguey, Bounded interpolation between lattices, J. Phys. A: Math. Gen., 24 (1991), 461-475.  doi: 10.1088/0305-4470/24/2/019.  Google Scholar

[8]

M. Duneau and C. Oguey, Displacive transformations and quasicrystalline symmetries, J. Phys. France, 51 (1990), 5-19.  doi: 10.1051/jphys:019900051010500.  Google Scholar

[9]

N. P. Frank and L. Sadun, Fusion: A general framework for hierarchical tilings of $ \mathbb{R}^d$, Geom. Dedicata, 171 (2014), 149-186.  doi: 10.1007/s10711-013-9893-7.  Google Scholar

[10]

D. Frettlöh and A. Garber, Pisot substitution sequences, one dimensional cut–and–project sets and bounded remainder sets with fractal boundary, Indag. Math., 29 (2018), 1114-1130.  doi: 10.1016/j.indag.2018.05.012.  Google Scholar

[11]

D. Frettlöh, Y. Smilansky and Y. Solomon, Bounded displacement non-equivalence in substitution tilings, J. Combin. Theory Ser. A, 177 (2021), 105326.  doi: 10.1016/j.jcta.2020.105326.  Google Scholar

[12]

A. I. Garber, On equivalence classes of separated nets (in Russian), Modeling and Analysis of Information Systems, 16 (2009), 109-118.   Google Scholar

[13]

S. Grepstad and N. Lev, Sets of bounded discrepancy for multi–dimensional irrational rotation, Geom. Funct. Anal., 25 (2015), 87-133.  doi: 10.1007/s00039-015-0313-z.  Google Scholar

[14]

A. Haynes, Equivalence classes of codimension one cut–and–project nets, Ergodic Theory Dyn. Syst., 36 (2016), 816-831.  doi: 10.1017/etds.2014.90.  Google Scholar

[15]

A. Haynes and H. Koivusalo, Constructing bounded remainder sets and cut–and–project sets which are bounded distance to lattices, Israel J. Math., 212 (2016), 189-201.  doi: 10.1007/s11856-016-1283-z.  Google Scholar

[16]

A. Haynes, M. Kelly and H. Koivusalo, Constructing bounded remainder sets and cut–and–project sets which are bounded distance to lattices Ⅱ, Indag. Math., 28 (2017), 138-144.  doi: 10.1016/j.indag.2016.11.010.  Google Scholar

[17]

C. Holton and L. Zamboni, Geometric realization of substitutions, Bull. Soc. Math. France, 126 (1998), 149-179.  doi: 10.24033/bsmf.2324.  Google Scholar

[18]

J. Kellendonk and I. F. T. Putnam, $C^*$-algebras, and $K$-theory, Directions in Mathematical Quasicrystals, CRM Monogr. Ser., 13, Amer. Math. Soc., Providence, RI, 2000,177–206.  Google Scholar

[19]

M. Laczkovich, Uniformly spread discrete sets in $ \mathbb{R}^d$, J. London Math. Soc., 46 (1992), 39-57.  doi: 10.1112/jlms/s2-46.1.39.  Google Scholar

[20]

J. C. Lagarias and P. A. B. Pleasants, Repetitive Delone sets and quasicrystals, Ergodic Theory Dynam. Systems, 23 (2003), 831-867.  doi: 10.1017/S0143385702001566.  Google Scholar

[21]

R. Rado, Factorization of even graphs, Quart. J. Math. Oxford, 20 (1949), 95-104.  doi: 10.1093/qmath/os-20.1.95.  Google Scholar

[22]

L. Sadun, Topology of Tiling Spaces, University Lecture Series, 46, American Mathematical Society, Providence, RI, 2008. doi: 10.1090/ulect/046.  Google Scholar

[23]

Y. Smilansky and Y. Solomon, A dichotomy for bounded displacement and Chabauty-Fell convergence of discrete sets, arXiv: 2011.00106, to appear In Erg. Th. and Dyn. Syst.. Google Scholar

[24]

Y. Solomon, Substitution tilings and separated nets with similarities to the integer lattice, Israel J. Math., 181 (2011), 445-460.  doi: 10.1007/s11856-011-0018-4.  Google Scholar

[25]

Y. Solomon, A simple condition for bounded displacement, J. Math. Anal. Appl., 414 (2014), 134-148.  doi: 10.1016/j.jmaa.2013.12.050.  Google Scholar

[26]

Y. Solomon, Continuously many bounded displacement non-equivalences in substitution tiling spaces, J. Math. Anal. Appl., 492 (2020), 124426.  doi: 10.1016/j.jmaa.2020.124426.  Google Scholar

show all references

References:
[1]

J. Aliste-Prieto, D. Coronel and J. M. Gambaudo, Linearly repetitive Delone sets are rectifiable, Ann. Inst. H. Poincaré Anal. Non Lineáire, 30 (2013), 275-290.  doi: 10.1016/j.anihpc.2012.07.006.  Google Scholar

[2]

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

[3] M. Baake and U. Grimm, Aperiodic Order, Cambridge Univ. Press, Cambridge, 2013.   Google Scholar
[4]

K. Bezdek, From the Kneser–Poulsen conjecture to ball–polyhedra, European J. Combin., 29 (2008), 1820-1830.  doi: 10.1016/j.ejc.2008.01.011.  Google Scholar

[5]

B. Csikós, A Schläfli–type formula for polytopes with curved faces and its application to the Kneser–Poulsen conjecture, Monatsh. Math, 147 (2006), 273-292.  doi: 10.1007/s00605-005-0363-7.  Google Scholar

[6]

W. A. Deuber, M. Simonovits and V. T. Sós, A note on paradoxical metric spaces, Stud. Sci. Math. Hungar., 30 (1995), 17-23.   Google Scholar

[7]

M. Duneau and C. Oguey, Bounded interpolation between lattices, J. Phys. A: Math. Gen., 24 (1991), 461-475.  doi: 10.1088/0305-4470/24/2/019.  Google Scholar

[8]

M. Duneau and C. Oguey, Displacive transformations and quasicrystalline symmetries, J. Phys. France, 51 (1990), 5-19.  doi: 10.1051/jphys:019900051010500.  Google Scholar

[9]

N. P. Frank and L. Sadun, Fusion: A general framework for hierarchical tilings of $ \mathbb{R}^d$, Geom. Dedicata, 171 (2014), 149-186.  doi: 10.1007/s10711-013-9893-7.  Google Scholar

[10]

D. Frettlöh and A. Garber, Pisot substitution sequences, one dimensional cut–and–project sets and bounded remainder sets with fractal boundary, Indag. Math., 29 (2018), 1114-1130.  doi: 10.1016/j.indag.2018.05.012.  Google Scholar

[11]

D. Frettlöh, Y. Smilansky and Y. Solomon, Bounded displacement non-equivalence in substitution tilings, J. Combin. Theory Ser. A, 177 (2021), 105326.  doi: 10.1016/j.jcta.2020.105326.  Google Scholar

[12]

A. I. Garber, On equivalence classes of separated nets (in Russian), Modeling and Analysis of Information Systems, 16 (2009), 109-118.   Google Scholar

[13]

S. Grepstad and N. Lev, Sets of bounded discrepancy for multi–dimensional irrational rotation, Geom. Funct. Anal., 25 (2015), 87-133.  doi: 10.1007/s00039-015-0313-z.  Google Scholar

[14]

A. Haynes, Equivalence classes of codimension one cut–and–project nets, Ergodic Theory Dyn. Syst., 36 (2016), 816-831.  doi: 10.1017/etds.2014.90.  Google Scholar

[15]

A. Haynes and H. Koivusalo, Constructing bounded remainder sets and cut–and–project sets which are bounded distance to lattices, Israel J. Math., 212 (2016), 189-201.  doi: 10.1007/s11856-016-1283-z.  Google Scholar

[16]

A. Haynes, M. Kelly and H. Koivusalo, Constructing bounded remainder sets and cut–and–project sets which are bounded distance to lattices Ⅱ, Indag. Math., 28 (2017), 138-144.  doi: 10.1016/j.indag.2016.11.010.  Google Scholar

[17]

C. Holton and L. Zamboni, Geometric realization of substitutions, Bull. Soc. Math. France, 126 (1998), 149-179.  doi: 10.24033/bsmf.2324.  Google Scholar

[18]

J. Kellendonk and I. F. T. Putnam, $C^*$-algebras, and $K$-theory, Directions in Mathematical Quasicrystals, CRM Monogr. Ser., 13, Amer. Math. Soc., Providence, RI, 2000,177–206.  Google Scholar

[19]

M. Laczkovich, Uniformly spread discrete sets in $ \mathbb{R}^d$, J. London Math. Soc., 46 (1992), 39-57.  doi: 10.1112/jlms/s2-46.1.39.  Google Scholar

[20]

J. C. Lagarias and P. A. B. Pleasants, Repetitive Delone sets and quasicrystals, Ergodic Theory Dynam. Systems, 23 (2003), 831-867.  doi: 10.1017/S0143385702001566.  Google Scholar

[21]

R. Rado, Factorization of even graphs, Quart. J. Math. Oxford, 20 (1949), 95-104.  doi: 10.1093/qmath/os-20.1.95.  Google Scholar

[22]

L. Sadun, Topology of Tiling Spaces, University Lecture Series, 46, American Mathematical Society, Providence, RI, 2008. doi: 10.1090/ulect/046.  Google Scholar

[23]

Y. Smilansky and Y. Solomon, A dichotomy for bounded displacement and Chabauty-Fell convergence of discrete sets, arXiv: 2011.00106, to appear In Erg. Th. and Dyn. Syst.. Google Scholar

[24]

Y. Solomon, Substitution tilings and separated nets with similarities to the integer lattice, Israel J. Math., 181 (2011), 445-460.  doi: 10.1007/s11856-011-0018-4.  Google Scholar

[25]

Y. Solomon, A simple condition for bounded displacement, J. Math. Anal. Appl., 414 (2014), 134-148.  doi: 10.1016/j.jmaa.2013.12.050.  Google Scholar

[26]

Y. Solomon, Continuously many bounded displacement non-equivalences in substitution tiling spaces, J. Math. Anal. Appl., 492 (2020), 124426.  doi: 10.1016/j.jmaa.2020.124426.  Google Scholar

[1]

Jeong-Yup Lee, Boris Solomyak. On substitution tilings and Delone sets without finite local complexity. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 3149-3177. doi: 10.3934/dcds.2019130
[2]

Michael Baake, Daniel Lenz. Spectral notions of aperiodic order. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 161-190. doi: 10.3934/dcdss.2017009
[3]

Steffen Klassert, Daniel Lenz, Peter Stollmann. Delone measures of finite local complexity and applications to spectral theory of one-dimensional continuum models of quasicrystals. Discrete & Continuous Dynamical Systems, 2011, 29 (4) : 1553-1571. doi: 10.3934/dcds.2011.29.1553
[4]

María Isabel Cortez, Samuel Petite. Realization of big centralizers of minimal aperiodic actions on the Cantor set. Discrete & Continuous Dynamical Systems, 2020, 40 (5) : 2891-2901. doi: 10.3934/dcds.2020153
[5]

Mariusz Lemańczyk, Clemens Müllner. Automatic sequences are orthogonal to aperiodic multiplicative functions. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6877-6918. doi: 10.3934/dcds.2020260
[6]

Xing Liu, Daiyuan Peng. Frequency hopping sequences with optimal aperiodic Hamming correlation by interleaving techniques. Advances in Mathematics of Communications, 2017, 11 (1) : 151-159. doi: 10.3934/amc.2017009
[7]

Alexander Gomilko, Mariusz Lemańczyk, Thierry de la Rue. On Furstenberg systems of aperiodic multiplicative functions of Matomäki, Radziwiłł, and Tao. Journal of Modern Dynamics, 2021, 17: 529-555. doi: 10.3934/jmd.2021018
[8]

Somphong Jitman, Ekkasit Sangwisut. The average dimension of the Hermitian hull of constacyclic codes over finite fields of square order. Advances in Mathematics of Communications, 2018, 12 (3) : 451-463. doi: 10.3934/amc.2018027
[9]

Natalie Priebe Frank, Lorenzo Sadun. Topology of some tiling spaces without finite local complexity. Discrete & Continuous Dynamical Systems, 2009, 23 (3) : 847-865. doi: 10.3934/dcds.2009.23.847
[10]

Yuan Guo, Xiaofei Gao, Desheng Li. Structure of the set of bounded solutions for a class of nonautonomous second order differential equations. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1607-1616. doi: 10.3934/cpaa.2010.9.1607
[11]

Alessandro Ferriero, Nicola Fusco. A note on the convex hull of sets of finite perimeter in the plane. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 103-108. doi: 10.3934/dcdsb.2009.11.103
[12]

Luis Barreira, Liviu Horia Popescu, Claudia Valls. Generalized exponential behavior and topological equivalence. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3023-3042. doi: 10.3934/dcdsb.2017161
[13]

Andres del Junco, Daniel J. Rudolph, Benjamin Weiss. Measured topological orbit and Kakutani equivalence. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 221-238. doi: 10.3934/dcdss.2009.2.221
[14]

Nam Yul Yu. A Fourier transform approach for improving the Levenshtein's lower bound on aperiodic correlation of binary sequences. Advances in Mathematics of Communications, 2014, 8 (2) : 209-222. doi: 10.3934/amc.2014.8.209
[15]

Xing Liu, Daiyuan Peng. Sets of frequency hopping sequences under aperiodic Hamming correlation: Upper bound and optimal constructions. Advances in Mathematics of Communications, 2014, 8 (3) : 359-373. doi: 10.3934/amc.2014.8.359
[16]

Yong Xia. Convex hull of the orthogonal similarity set with applications in quadratic assignment problems. Journal of Industrial & Management Optimization, 2013, 9 (3) : 689-701. doi: 10.3934/jimo.2013.9.689
[17]

Giuseppe Buttazzo, Luigi De Pascale, Ilaria Fragalà. Topological equivalence of some variational problems involving distances. Discrete & Continuous Dynamical Systems, 2001, 7 (2) : 247-258. doi: 10.3934/dcds.2001.7.247
[18]

Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, 2021, 15 (2) : 315-338. doi: 10.3934/ipi.2020070
[19]

Stefano Galatolo. Global and local complexity in weakly chaotic dynamical systems. Discrete & Continuous Dynamical Systems, 2003, 9 (6) : 1607-1624. doi: 10.3934/dcds.2003.9.1607
[20]

Álvaro Castañeda, Pablo González, Gonzalo Robledo. Topological Equivalence of nonautonomous difference equations with a family of dichotomies on the half line. Communications on Pure & Applied Analysis, 2021, 20 (2) : 511-532. doi: 10.3934/cpaa.2020278

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (45)
  • HTML views (42)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy