-
Previous Article
Approximations for Gibbs states of arbitrary Hölder potentials on hyperbolic folded sets
- DCDS Home
- This Issue
-
Next Article
Phase portraits of predator--prey systems with harvesting rates
Pisot family self-affine tilings, discrete spectrum, and the Meyer property
1. | Dept. of Math. Edu., Kwandong University, 522 Naegok-dong, Gangneung, Gangwon 210-701, South Korea |
2. | Box 354350, Department of Mathematics, University of Washington, Seattle WA 98195, United States |
References:
[1] |
J. Aczél, "Lectures on Functional Equations and Their Applications,'', Mathematics in Science and Engineering, (1966). Google Scholar |
[2] |
S. Akiyama and J.-Y. Lee, Algorithm for determining pure pointedness of self-affine tilings,, Adv. Math., 226 (2011), 2855.
doi: 10.1016/j.aim.2010.07.019. |
[3] |
J. Andersen and I. Putnam, Topological invariants for substitution tilings and their associated $C^\mathbf{star}$-algebras,, Ergodic Theory Dynam. Systems, 18 (1998), 509.
doi: 10.1017/S0143385798100457. |
[4] |
M. Baake and D. Lenz, Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra,, Ergodic Theory Dynam. Systems, 24 (2004), 1867.
doi: 10.1017/S0143385704000318. |
[5] |
M. Baake, D. Lenz and R. V. Moody, Characterization of model sets by dynamical systems,, Ergodic Theory Dynam. Systems, 27 (2007), 341.
doi: 10.1017/S0143385706000800. |
[6] |
R. Benedetti and J.-M. Gambaudo, On the dynamics of $\mathbb G$-solenoids. Applications to Delone sets,, Ergodic Theory Dynam. Systems, 23 (2003), 673.
doi: 10.1017/S0143385702001578. |
[7] |
A. Clark and L. Sadun, When shape matters: Deformations of tiling spaces,, Ergodic Theory Dynam. Systems, 26 (2006), 69.
doi: 10.1017/S0143385705000623. |
[8] |
L. Danzer, Inflation species of planar tilings which are not of locally finite complexity,, Proc. Steklov Inst. Math., 239 (2002), 108.
|
[9] |
S. Dworkin, Spectral theory and $x$-ray diffraction,, J. Math. Phys., 34 (1993), 2965.
doi: 10.1063/1.530108. |
[10] |
N. P. Frank, A primer of substitution tilings of the Euclidean plane,, Expo. Math., 26 (2008), 295.
doi: 10.1016/j.exmath.2008.02.001. |
[11] |
N. P. Frank and E. A. Robinson, Jr., Generalized $\beta$-expansions, substitution tilings, and local finiteness,, Trans. Amer. Math. Soc., 360 (2008), 1163.
doi: 10.1090/S0002-9947-07-04527-8. |
[12] |
J.-M. Gambaudo, A note on tilings and translation surfaces,, Ergodic Theory Dynam. Systems, 26 (2006), 179.
doi: 10.1017/S0143385705000404. |
[13] |
J.-B. Gouéré, Quasicrystals and almost periodicity,, Comm. Math. Phys., 255 (2005), 655.
doi: 10.1007/s00220-004-1271-8. |
[14] |
M. Hirsch and S. Smale, "Differential Equations, Dynamical Systems, and Linear Algebra,'', Pure and Applied Mathematics, (1974).
|
[15] |
C. Holton, C. Radin and L. Sadun, Conjugacies for tiling dynamical systems,, Comm. Math. Phys., 254 (2005), 343.
doi: 10.1007/s00220-004-1195-3. |
[16] |
J. Kellendonk, Pattern equivariant functions, deformations and equivalence of tiling spaces,, Ergodic Theory Dynam. Systems, 28 (2008), 1153.
doi: 10.1017/S014338570700065X. |
[17] |
R. Kenyon, Self-replicating tilings,, in, (1991), 239.
|
[18] |
R. Kenyon, Inflationary tilings with a similarity structure,, Comment. Math. Helv., 69 (1994), 169.
doi: 10.1007/BF02564481. |
[19] |
R. Kenyon, The construction of self-similar tilings,, Geom. Funct. Anal., 6 (1996), 471.
doi: 10.1007/BF02249260. |
[20] |
R. Kenyon, "Self-Similar Tilings,'', Ph.D Thesis, (1990).
|
[21] |
R. Kenyon and B. Solomyak, On the characterization of expansion maps for self-affine tilings,, Discrete Comput. Geom., 43 (2010), 577.
|
[22] |
J. C. Lagarias, Mathematical quasicrystals and the problem of diffraction,, in, (2000), 61.
|
[23] |
J. C. Lagarias and Y. Wang, Substitution Delone sets,, Discrete Comput. Geom., 29 (2003), 175.
doi: 10.1007/s00454-002-2820-6. |
[24] |
J.-Y. Lee, Substitution Delone sets with pure point spectrum are inter-model sets,, J. Geom. Phys., 57 (2007), 2263.
doi: 10.1016/j.geomphys.2007.07.003. |
[25] |
J.-Y. Lee, R. V. Moody and B. Solomyak, Pure point dynamical and diffraction spectra,, Ann. Henri Poincaré, 3 (2002), 1003.
doi: 10.1007/s00023-002-8646-1. |
[26] |
J.-Y. Lee, R. V. Moody and B. Solomyak, Consequences of pure point diffraction spectra for multiset substitution systems,, Discrete Comp. Geom., 29 (2003), 525.
doi: 10.1007/s00454-003-0781-z. |
[27] |
J.-Y Lee and B. Solomyak, Pure point diffractive substitution Delone sets have the Meyer property,, Discrete Comput. Geom., 39 (2008), 319.
doi: 10.1007/s00454-008-9054-1. |
[28] |
D. Lind, The entropies of topological Markov shifts and a related class of algebraic integers,, Ergodic Theory Dynam. Systems, 4 (1984), 283. Google Scholar |
[29] |
C. Mauduit, Caractérisation des ensembles normaux substitutifs,, Invent. Math., 95 (1989), 133.
doi: 10.1007/BF01394146. |
[30] |
R. V. Moody, Meyer sets and their duals,, in, (1997), 403.
|
[31] |
S. Mozes, Tilings, substitution systems and dynamical systems generated by them,, J. Anal. Math., 53 (1989), 139.
doi: 10.1007/BF02793412. |
[32] |
K. Petersen, Factor maps between tiling dynamical systems,, Forum Math., 11 (1999), 503.
doi: 10.1515/form.1999.011. |
[33] |
B. Praggastis, Numeration systems and Markov partitions from self-similar tilings,, Trans. Amer. Math. Soc., 351 (1999), 3315.
doi: 10.1090/S0002-9947-99-02360-0. |
[34] |
C. Radin, The pinwheel tilings of the plane,, Annals of Math., 139 (1994), 661.
doi: 10.2307/2118575. |
[35] |
E. A. Robinson, Symbolic dynamics and tilings of $\mathbbR^d$, in "Symbolic Dynamics and its Applications,", 81-119, 60 (2004), 81.
|
[36] |
L. Sadun, Some generalizations of the Pinwheel tiling,, Discrete Comput. Geom., 20 (1998), 79.
doi: 10.1007/PL00009379. |
[37] |
L. Sadun, "Topology of Tiling Spaces,'', University Lecture Series, 46 (2008).
|
[38] |
B. Solomyak, Corrections to: "Dynamics of self-similar tilings",, [Ergodic Theory Dynam. Systems 17 (1997), 17 (1997), 695.
doi: 10.1017/S014338579917161X. |
[39] |
B. Solomyak, Nonperiodicity implies unique composition for self-similar translationally finite tilings,, Discrete Comput. Geom., 20 (1998), 265.
doi: 10.1007/PL00009386. |
[40] |
B. Solomyak, Eigenfunctions for substitution tiling systems,, in, 49 (2005), 433.
|
[41] |
B. Solomyak, Tilings and dynamics,, Lecture Notes, (2006), 8. Google Scholar |
[42] |
E. M. Stein, "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals,'', With the assistance of Timothy S. Murphy, 43 (1993).
|
[43] |
W. Thurston, "Groups, Tilings, and Finite State Automata,'', AMS lecture notes, (1989). Google Scholar |
[44] |
T. Vijayaraghavan, On the fractional parts of the powers of a number. II,, Proc. Cambridge Philos. Soc., 37 (1941), 349.
doi: 10.1017/S0305004100017989. |
show all references
References:
[1] |
J. Aczél, "Lectures on Functional Equations and Their Applications,'', Mathematics in Science and Engineering, (1966). Google Scholar |
[2] |
S. Akiyama and J.-Y. Lee, Algorithm for determining pure pointedness of self-affine tilings,, Adv. Math., 226 (2011), 2855.
doi: 10.1016/j.aim.2010.07.019. |
[3] |
J. Andersen and I. Putnam, Topological invariants for substitution tilings and their associated $C^\mathbf{star}$-algebras,, Ergodic Theory Dynam. Systems, 18 (1998), 509.
doi: 10.1017/S0143385798100457. |
[4] |
M. Baake and D. Lenz, Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra,, Ergodic Theory Dynam. Systems, 24 (2004), 1867.
doi: 10.1017/S0143385704000318. |
[5] |
M. Baake, D. Lenz and R. V. Moody, Characterization of model sets by dynamical systems,, Ergodic Theory Dynam. Systems, 27 (2007), 341.
doi: 10.1017/S0143385706000800. |
[6] |
R. Benedetti and J.-M. Gambaudo, On the dynamics of $\mathbb G$-solenoids. Applications to Delone sets,, Ergodic Theory Dynam. Systems, 23 (2003), 673.
doi: 10.1017/S0143385702001578. |
[7] |
A. Clark and L. Sadun, When shape matters: Deformations of tiling spaces,, Ergodic Theory Dynam. Systems, 26 (2006), 69.
doi: 10.1017/S0143385705000623. |
[8] |
L. Danzer, Inflation species of planar tilings which are not of locally finite complexity,, Proc. Steklov Inst. Math., 239 (2002), 108.
|
[9] |
S. Dworkin, Spectral theory and $x$-ray diffraction,, J. Math. Phys., 34 (1993), 2965.
doi: 10.1063/1.530108. |
[10] |
N. P. Frank, A primer of substitution tilings of the Euclidean plane,, Expo. Math., 26 (2008), 295.
doi: 10.1016/j.exmath.2008.02.001. |
[11] |
N. P. Frank and E. A. Robinson, Jr., Generalized $\beta$-expansions, substitution tilings, and local finiteness,, Trans. Amer. Math. Soc., 360 (2008), 1163.
doi: 10.1090/S0002-9947-07-04527-8. |
[12] |
J.-M. Gambaudo, A note on tilings and translation surfaces,, Ergodic Theory Dynam. Systems, 26 (2006), 179.
doi: 10.1017/S0143385705000404. |
[13] |
J.-B. Gouéré, Quasicrystals and almost periodicity,, Comm. Math. Phys., 255 (2005), 655.
doi: 10.1007/s00220-004-1271-8. |
[14] |
M. Hirsch and S. Smale, "Differential Equations, Dynamical Systems, and Linear Algebra,'', Pure and Applied Mathematics, (1974).
|
[15] |
C. Holton, C. Radin and L. Sadun, Conjugacies for tiling dynamical systems,, Comm. Math. Phys., 254 (2005), 343.
doi: 10.1007/s00220-004-1195-3. |
[16] |
J. Kellendonk, Pattern equivariant functions, deformations and equivalence of tiling spaces,, Ergodic Theory Dynam. Systems, 28 (2008), 1153.
doi: 10.1017/S014338570700065X. |
[17] |
R. Kenyon, Self-replicating tilings,, in, (1991), 239.
|
[18] |
R. Kenyon, Inflationary tilings with a similarity structure,, Comment. Math. Helv., 69 (1994), 169.
doi: 10.1007/BF02564481. |
[19] |
R. Kenyon, The construction of self-similar tilings,, Geom. Funct. Anal., 6 (1996), 471.
doi: 10.1007/BF02249260. |
[20] |
R. Kenyon, "Self-Similar Tilings,'', Ph.D Thesis, (1990).
|
[21] |
R. Kenyon and B. Solomyak, On the characterization of expansion maps for self-affine tilings,, Discrete Comput. Geom., 43 (2010), 577.
|
[22] |
J. C. Lagarias, Mathematical quasicrystals and the problem of diffraction,, in, (2000), 61.
|
[23] |
J. C. Lagarias and Y. Wang, Substitution Delone sets,, Discrete Comput. Geom., 29 (2003), 175.
doi: 10.1007/s00454-002-2820-6. |
[24] |
J.-Y. Lee, Substitution Delone sets with pure point spectrum are inter-model sets,, J. Geom. Phys., 57 (2007), 2263.
doi: 10.1016/j.geomphys.2007.07.003. |
[25] |
J.-Y. Lee, R. V. Moody and B. Solomyak, Pure point dynamical and diffraction spectra,, Ann. Henri Poincaré, 3 (2002), 1003.
doi: 10.1007/s00023-002-8646-1. |
[26] |
J.-Y. Lee, R. V. Moody and B. Solomyak, Consequences of pure point diffraction spectra for multiset substitution systems,, Discrete Comp. Geom., 29 (2003), 525.
doi: 10.1007/s00454-003-0781-z. |
[27] |
J.-Y Lee and B. Solomyak, Pure point diffractive substitution Delone sets have the Meyer property,, Discrete Comput. Geom., 39 (2008), 319.
doi: 10.1007/s00454-008-9054-1. |
[28] |
D. Lind, The entropies of topological Markov shifts and a related class of algebraic integers,, Ergodic Theory Dynam. Systems, 4 (1984), 283. Google Scholar |
[29] |
C. Mauduit, Caractérisation des ensembles normaux substitutifs,, Invent. Math., 95 (1989), 133.
doi: 10.1007/BF01394146. |
[30] |
R. V. Moody, Meyer sets and their duals,, in, (1997), 403.
|
[31] |
S. Mozes, Tilings, substitution systems and dynamical systems generated by them,, J. Anal. Math., 53 (1989), 139.
doi: 10.1007/BF02793412. |
[32] |
K. Petersen, Factor maps between tiling dynamical systems,, Forum Math., 11 (1999), 503.
doi: 10.1515/form.1999.011. |
[33] |
B. Praggastis, Numeration systems and Markov partitions from self-similar tilings,, Trans. Amer. Math. Soc., 351 (1999), 3315.
doi: 10.1090/S0002-9947-99-02360-0. |
[34] |
C. Radin, The pinwheel tilings of the plane,, Annals of Math., 139 (1994), 661.
doi: 10.2307/2118575. |
[35] |
E. A. Robinson, Symbolic dynamics and tilings of $\mathbbR^d$, in "Symbolic Dynamics and its Applications,", 81-119, 60 (2004), 81.
|
[36] |
L. Sadun, Some generalizations of the Pinwheel tiling,, Discrete Comput. Geom., 20 (1998), 79.
doi: 10.1007/PL00009379. |
[37] |
L. Sadun, "Topology of Tiling Spaces,'', University Lecture Series, 46 (2008).
|
[38] |
B. Solomyak, Corrections to: "Dynamics of self-similar tilings",, [Ergodic Theory Dynam. Systems 17 (1997), 17 (1997), 695.
doi: 10.1017/S014338579917161X. |
[39] |
B. Solomyak, Nonperiodicity implies unique composition for self-similar translationally finite tilings,, Discrete Comput. Geom., 20 (1998), 265.
doi: 10.1007/PL00009386. |
[40] |
B. Solomyak, Eigenfunctions for substitution tiling systems,, in, 49 (2005), 433.
|
[41] |
B. Solomyak, Tilings and dynamics,, Lecture Notes, (2006), 8. Google Scholar |
[42] |
E. M. Stein, "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals,'', With the assistance of Timothy S. Murphy, 43 (1993).
|
[43] |
W. Thurston, "Groups, Tilings, and Finite State Automata,'', AMS lecture notes, (1989). Google Scholar |
[44] |
T. Vijayaraghavan, On the fractional parts of the powers of a number. II,, Proc. Cambridge Philos. Soc., 37 (1941), 349.
doi: 10.1017/S0305004100017989. |
[1] |
Evan Greif, Daniel Kaplan, Robert S. Strichartz, Samuel C. Wiese. Spectrum of the Laplacian on regular polyhedra. Communications on Pure & Applied Analysis, 2021, 20 (1) : 193-214. doi: 10.3934/cpaa.2020263 |
[2] |
Bing Gao, Rui Gao. On fair entropy of the tent family. Discrete & Continuous Dynamical Systems - A, 2021 doi: 10.3934/dcds.2021017 |
[3] |
Jesús A. Álvarez López, Ramón Barral Lijó, John Hunton, Hiraku Nozawa, John R. Parker. Chaotic Delone sets. Discrete & Continuous Dynamical Systems - A, 2021 doi: 10.3934/dcds.2021016 |
[4] |
Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217 |
[5] |
Stefan Ruschel, Serhiy Yanchuk. The spectrum of delay differential equations with multiple hierarchical large delays. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 151-175. doi: 10.3934/dcdss.2020321 |
[6] |
Claudio Bonanno, Marco Lenci. Pomeau-Manneville maps are global-local mixing. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1051-1069. doi: 10.3934/dcds.2020309 |
[7] |
Xinyu Mei, Yangmin Xiong, Chunyou Sun. Pullback attractor for a weakly damped wave equation with sup-cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 569-600. doi: 10.3934/dcds.2020270 |
[8] |
Bopeng Rao, Zhuangyi Liu. A spectral approach to the indirect boundary control of a system of weakly coupled wave equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 399-414. doi: 10.3934/dcds.2009.23.399 |
[9] |
Riccarda Rossi, Ulisse Stefanelli, Marita Thomas. Rate-independent evolution of sets. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 89-119. doi: 10.3934/dcdss.2020304 |
[10] |
Buddhadev Pal, Pankaj Kumar. A family of multiply warped product semi-Riemannian Einstein metrics. Journal of Geometric Mechanics, 2020, 12 (4) : 553-562. doi: 10.3934/jgm.2020017 |
[11] |
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, , () : -. doi: 10.3934/cpaa.2020278 |
[12] |
Wolfgang Riedl, Robert Baier, Matthias Gerdts. Optimization-based subdivision algorithm for reachable sets. Journal of Computational Dynamics, 2021, 8 (1) : 99-130. doi: 10.3934/jcd.2021005 |
[13] |
Vito Napolitano, Ferdinando Zullo. Codes with few weights arising from linear sets. Advances in Mathematics of Communications, 2020 doi: 10.3934/amc.2020129 |
[14] |
Lisa Hernandez Lucas. Properties of sets of Subspaces with Constant Intersection Dimension. Advances in Mathematics of Communications, 2021, 15 (1) : 191-206. doi: 10.3934/amc.2020052 |
[15] |
Shin-Ichiro Ei, Hiroshi Ishii. The motion of weakly interacting localized patterns for reaction-diffusion systems with nonlocal effect. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 173-190. doi: 10.3934/dcdsb.2020329 |
[16] |
Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168 |
[17] |
Parikshit Upadhyaya, Elias Jarlebring, Emanuel H. Rubensson. A density matrix approach to the convergence of the self-consistent field iteration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 99-115. doi: 10.3934/naco.2020018 |
[18] |
Fanni M. Sélley. A self-consistent dynamical system with multiple absolutely continuous invariant measures. Journal of Computational Dynamics, 2021, 8 (1) : 9-32. doi: 10.3934/jcd.2021002 |
[19] |
Meng Ding, Ting-Zhu Huang, Xi-Le Zhao, Michael K. Ng, Tian-Hui Ma. Tensor train rank minimization with nonlocal self-similarity for tensor completion. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021001 |
[20] |
Shin-Ichiro Ei, Masayasu Mimura, Tomoyuki Miyaji. Reflection of a self-propelling rigid disk from a boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 803-817. doi: 10.3934/dcdss.2020229 |
2019 Impact Factor: 1.338
Tools
Metrics
Other articles
by authors
[Back to Top]