-
Previous Article
Particle approximation of the one dimensional Keller-Segel equation, stability and rigidity of the blow-up
- DCDS Home
- This Issue
-
Next Article
An improved Hardy inequality for a nonlocal operator
Pure discrete spectrum for a class of one-dimensional substitution tiling systems
1. | Department of Mathematics, Montana State University, Bozeman, MT 59717 |
References:
[1] |
S. Akiyama, M. Barge, V. Berthé, J.-Y. Lee and A. Siegel, On the Pisot Conjecture, in Mathematics of Aperiodic Order (eds. J. Kellendonk, D. Lenz and J. Savinien), Progress in Mathematics, 309, 2015, 33-72. |
[2] |
S. Akiyama and J.-Y. Lee, Algorithm for determining pure pointedness of self-affine tilings, Adv. Math., 226, (2011), 2855-2883.
doi: 10.1016/j.aim.2010.07.019. |
[3] |
S. Akiyama and J.-Y. Lee, Computation of pure discrete spectrum of self-affine tilings,, preprint., ().
|
[4] |
J. E. Anderson and I. F. Putnam, Topological invariants for substitution tilings and their associated $C^*$-algebras, Ergodic Theory & Dynamical Systems, 18 (1998), 509-537.
doi: 10.1017/S0143385798100457. |
[5] |
P. Arnoux and S. Ito, Pisot Substitutions and Rauzy fractals, Bull. Belg. Math Soc., 8 (2001), 181-207. |
[6] |
A. Avila and V. Delecroix, Some monoids of Pisot matrices, preprint,, , ().
|
[7] |
V. Baker, M. Barge and J. Kwapisz, Geometric realization and coincidence for reducible non-unimodular Pisot tiling spaces with an application to $\beta$-shifts, Ann. Inst. Fourier (Grenoble), 56 (2006), 2213-2248.
doi: 10.5802/aif.2238. |
[8] |
M. Barge, Factors of Pisot tiling spaces and the coincidence rank conjecture,, , ().
|
[9] |
M. Barge and B. Diamond, A complete invariant for the topology of one- dimensional substitution tiling spaces, Ergod. Th. & Dyn. Sys., 21 (2001), 1333-1358.
doi: 10.1017/S0143385701001638. |
[10] |
M. Barge and B. Diamond, Coincidence for substitutions of Pisot type, Bull. Soc. Math. France, 130 (2002), 619-626. |
[11] |
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. |
[12] |
M. Barge and J. Kwapisz, Geometric theory of unimodular Pisot substitutions, Amer J. Math., {\bf128} (2006), 1219-1282.
doi: 10.1353/ajm.2006.0037. |
[13] |
M. Barge, S. Štimac and R. F. Williams, Pure discrete spectrum in substitution tiling spaces, Disc. and Cont. Dynam. Sys. - A, 2 (2013), 579-597.
doi: 10.3934/dcds.2013.33.579. |
[14] |
V. Berthé, J. Bourdon, T. Jolivet and A. Siegel, A combinatorial approach to products of Pisot substitutions,, , ().
|
[15] |
V. Berthé, S. Ferenczi and L. Q. Zamboni, Interactions between dynamics, arihtmetics and combinatorics: The good, the bad, and the ugly, in Algebraic and Topological Dynamics, Contemp. Math., 385, Amer. Math. Soc., Providence, RI, 2005, 333-364.
doi: 10.1090/conm/385/07205. |
[16] |
V. Berthé, T. Jolivet and A. Siegel, Substitutive Arnoux-Rauzy substitutions have pure discrete spectrum, Unif. Distrib. Theory, 7 (2012), 173-197. |
[17] |
V. Berthé, Multidimensional Euclidean algorithms, numeration and substitutions, Integers, 11B (2011), Paper No. A2, 34pp. |
[18] |
A. Bertrand, Développements en base de Pisot et répartition modulo 1, C. R. Acad. Sci. Paris, 280 (1979), A1-A4. |
[19] |
J. Cassaigne and N. Chekhova, Fonctions de récurrence des suites d'Arnoux-Rauzy et réponse à une questionde Morse et Hedlund, Ann. Inst. Fourier (Grenoble), Numération, Pavages, Substitutions, 56 (2006), 2249-2270.
doi: 10.5802/aif.2239. |
[20] |
A. Clark and L. Sadun, When size matters: Subshifts and their related tiling spaces, Ergodic Theory Dynam. Systems, 23 (2003), 1043-1057.
doi: 10.1017/S0143385702001633. |
[21] |
E. Dubois, A. Farhane and R. Paysant-Le Roux, The Jacobi-Perron algorithm and Pisot numbers, Acta Arith., 111 (2004), 269-275.
doi: 10.4064/aa111-3-4. |
[22] |
H. Ei and S. Ito, Tilings from some non-irreducible, Pisot substitutions, Discrete Math. and Theo. Comp. Science, 7 (2005), 81-121. |
[23] |
M. Hollander and B. Solomyak, Two-symbol Pisot substitutions have pure discrete spectrum, Ergodic Theory & Dynamical Systems, 23 (2003), 533-540.
doi: 10.1017/S0143385702001384. |
[24] |
B. Mossé, Puissances de mots et reconnaissabilité des points fixes d'une substitution, Theoretical Computer Science, 99 (1992), 327-334.
doi: 10.1016/0304-3975(92)90357-L. |
[25] |
F. Schweiger, Multidimensional Continued Fractions, Oxford Science Publications, Oxford University Press, Oxford, 2000. |
[26] |
K. Schmidt, On periodic expansions of Pisot numbers and Salem numbers, Bull. London Math. Soc., 12 (1980), 269-278.
doi: 10.1112/blms/12.4.269. |
[27] |
N. Sidorov, Arithmetic dynamics, in Topics in Dynamics and Ergodic Theory, London Math. Soc. Lecture Note Ser., 310, Cambridge Univ. Press, Cambridge, 2003, 145-189.
doi: 10.1017/CBO9780511546716.010. |
[28] |
B. Solomyak, Dynamics of self-similar tilings, Ergod. Th. & Dynam. Sys., 17 (1997), 695-738.
doi: 10.1017/S0143385797084988. |
[29] |
B. Solomyak, Nonperiodicity implies unique composition for self-similar translationally finite tilings, Discrete Comput. Geometry, 20 (1998), 265-279.
doi: 10.1007/PL00009386. |
[30] |
P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York, 1982. |
show all references
References:
[1] |
S. Akiyama, M. Barge, V. Berthé, J.-Y. Lee and A. Siegel, On the Pisot Conjecture, in Mathematics of Aperiodic Order (eds. J. Kellendonk, D. Lenz and J. Savinien), Progress in Mathematics, 309, 2015, 33-72. |
[2] |
S. Akiyama and J.-Y. Lee, Algorithm for determining pure pointedness of self-affine tilings, Adv. Math., 226, (2011), 2855-2883.
doi: 10.1016/j.aim.2010.07.019. |
[3] |
S. Akiyama and J.-Y. Lee, Computation of pure discrete spectrum of self-affine tilings,, preprint., ().
|
[4] |
J. E. Anderson and I. F. Putnam, Topological invariants for substitution tilings and their associated $C^*$-algebras, Ergodic Theory & Dynamical Systems, 18 (1998), 509-537.
doi: 10.1017/S0143385798100457. |
[5] |
P. Arnoux and S. Ito, Pisot Substitutions and Rauzy fractals, Bull. Belg. Math Soc., 8 (2001), 181-207. |
[6] |
A. Avila and V. Delecroix, Some monoids of Pisot matrices, preprint,, , ().
|
[7] |
V. Baker, M. Barge and J. Kwapisz, Geometric realization and coincidence for reducible non-unimodular Pisot tiling spaces with an application to $\beta$-shifts, Ann. Inst. Fourier (Grenoble), 56 (2006), 2213-2248.
doi: 10.5802/aif.2238. |
[8] |
M. Barge, Factors of Pisot tiling spaces and the coincidence rank conjecture,, , ().
|
[9] |
M. Barge and B. Diamond, A complete invariant for the topology of one- dimensional substitution tiling spaces, Ergod. Th. & Dyn. Sys., 21 (2001), 1333-1358.
doi: 10.1017/S0143385701001638. |
[10] |
M. Barge and B. Diamond, Coincidence for substitutions of Pisot type, Bull. Soc. Math. France, 130 (2002), 619-626. |
[11] |
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. |
[12] |
M. Barge and J. Kwapisz, Geometric theory of unimodular Pisot substitutions, Amer J. Math., {\bf128} (2006), 1219-1282.
doi: 10.1353/ajm.2006.0037. |
[13] |
M. Barge, S. Štimac and R. F. Williams, Pure discrete spectrum in substitution tiling spaces, Disc. and Cont. Dynam. Sys. - A, 2 (2013), 579-597.
doi: 10.3934/dcds.2013.33.579. |
[14] |
V. Berthé, J. Bourdon, T. Jolivet and A. Siegel, A combinatorial approach to products of Pisot substitutions,, , ().
|
[15] |
V. Berthé, S. Ferenczi and L. Q. Zamboni, Interactions between dynamics, arihtmetics and combinatorics: The good, the bad, and the ugly, in Algebraic and Topological Dynamics, Contemp. Math., 385, Amer. Math. Soc., Providence, RI, 2005, 333-364.
doi: 10.1090/conm/385/07205. |
[16] |
V. Berthé, T. Jolivet and A. Siegel, Substitutive Arnoux-Rauzy substitutions have pure discrete spectrum, Unif. Distrib. Theory, 7 (2012), 173-197. |
[17] |
V. Berthé, Multidimensional Euclidean algorithms, numeration and substitutions, Integers, 11B (2011), Paper No. A2, 34pp. |
[18] |
A. Bertrand, Développements en base de Pisot et répartition modulo 1, C. R. Acad. Sci. Paris, 280 (1979), A1-A4. |
[19] |
J. Cassaigne and N. Chekhova, Fonctions de récurrence des suites d'Arnoux-Rauzy et réponse à une questionde Morse et Hedlund, Ann. Inst. Fourier (Grenoble), Numération, Pavages, Substitutions, 56 (2006), 2249-2270.
doi: 10.5802/aif.2239. |
[20] |
A. Clark and L. Sadun, When size matters: Subshifts and their related tiling spaces, Ergodic Theory Dynam. Systems, 23 (2003), 1043-1057.
doi: 10.1017/S0143385702001633. |
[21] |
E. Dubois, A. Farhane and R. Paysant-Le Roux, The Jacobi-Perron algorithm and Pisot numbers, Acta Arith., 111 (2004), 269-275.
doi: 10.4064/aa111-3-4. |
[22] |
H. Ei and S. Ito, Tilings from some non-irreducible, Pisot substitutions, Discrete Math. and Theo. Comp. Science, 7 (2005), 81-121. |
[23] |
M. Hollander and B. Solomyak, Two-symbol Pisot substitutions have pure discrete spectrum, Ergodic Theory & Dynamical Systems, 23 (2003), 533-540.
doi: 10.1017/S0143385702001384. |
[24] |
B. Mossé, Puissances de mots et reconnaissabilité des points fixes d'une substitution, Theoretical Computer Science, 99 (1992), 327-334.
doi: 10.1016/0304-3975(92)90357-L. |
[25] |
F. Schweiger, Multidimensional Continued Fractions, Oxford Science Publications, Oxford University Press, Oxford, 2000. |
[26] |
K. Schmidt, On periodic expansions of Pisot numbers and Salem numbers, Bull. London Math. Soc., 12 (1980), 269-278.
doi: 10.1112/blms/12.4.269. |
[27] |
N. Sidorov, Arithmetic dynamics, in Topics in Dynamics and Ergodic Theory, London Math. Soc. Lecture Note Ser., 310, Cambridge Univ. Press, Cambridge, 2003, 145-189.
doi: 10.1017/CBO9780511546716.010. |
[28] |
B. Solomyak, Dynamics of self-similar tilings, Ergod. Th. & Dynam. Sys., 17 (1997), 695-738.
doi: 10.1017/S0143385797084988. |
[29] |
B. Solomyak, Nonperiodicity implies unique composition for self-similar translationally finite tilings, Discrete Comput. Geometry, 20 (1998), 265-279.
doi: 10.1007/PL00009386. |
[30] |
P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York, 1982. |
[1] |
Marcy Barge, Sonja Štimac, R. F. Williams. Pure discrete spectrum in substitution tiling spaces. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 579-597. doi: 10.3934/dcds.2013.33.579 |
[2] |
Jeanette Olli. Endomorphisms of Sturmian systems and the discrete chair substitution tiling system. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 4173-4186. doi: 10.3934/dcds.2013.33.4173 |
[3] |
Rui Pacheco, Helder Vilarinho. Statistical stability for multi-substitution tiling spaces. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 4579-4594. doi: 10.3934/dcds.2013.33.4579 |
[4] |
Noriaki Kawaguchi. Maximal chain continuous factor. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5915-5942. doi: 10.3934/dcds.2021101 |
[5] |
Gerhard Keller. Maximal equicontinuous generic factors and weak model sets. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6855-6875. doi: 10.3934/dcds.2020132 |
[6] |
Nazar Arakelian, Saeed Tafazolian, Fernando Torres. On the spectrum for the genera of maximal curves over small fields. Advances in Mathematics of Communications, 2018, 12 (1) : 143-149. doi: 10.3934/amc.2018009 |
[7] |
Tao Yu, Guohua Zhang, Ruifeng Zhang. Discrete spectrum for amenable group actions. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5871-5886. doi: 10.3934/dcds.2021099 |
[8] |
Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 3043-3054. doi: 10.3934/dcdss.2020463 |
[9] |
Evelyn Herberg, Michael Hinze, Henrik Schumacher. Maximal discrete sparsity in parabolic optimal control with measures. Mathematical Control and Related Fields, 2020, 10 (4) : 735-759. doi: 10.3934/mcrf.2020018 |
[10] |
Kathryn Dabbs, Michael Kelly, Han Li. Effective equidistribution of translates of maximal horospherical measures in the space of lattices. Journal of Modern Dynamics, 2016, 10: 229-254. doi: 10.3934/jmd.2016.10.229 |
[11] |
Carlos Lizama, Marina Murillo-Arcila. Discrete maximal regularity for volterra equations and nonlocal time-stepping schemes. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 509-528. doi: 10.3934/dcds.2020020 |
[12] |
Bassam Fayad, A. Windsor. A dichotomy between discrete and continuous spectrum for a class of special flows over rotations. Journal of Modern Dynamics, 2007, 1 (1) : 107-122. doi: 10.3934/jmd.2007.1.107 |
[13] |
Nikolai Edeko. On the isomorphism problem for non-minimal transformations with discrete spectrum. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 6001-6021. doi: 10.3934/dcds.2019262 |
[14] |
Jeong-Yup Lee, Boris Solomyak. Pisot family self-affine tilings, discrete spectrum, and the Meyer property. Discrete and Continuous Dynamical Systems, 2012, 32 (3) : 935-959. doi: 10.3934/dcds.2012.32.935 |
[15] |
Wen Huang, Zhiren Wang, Guohua Zhang. Möbius disjointness for topological models of ergodic systems with discrete spectrum. Journal of Modern Dynamics, 2019, 14: 277-290. doi: 10.3934/jmd.2019010 |
[16] |
Alexander Vladimirov. Equicontinuous sweeping processes. Discrete and Continuous Dynamical Systems - B, 2013, 18 (2) : 565-573. doi: 10.3934/dcdsb.2013.18.565 |
[17] |
Yoshikazu Katayama, Colin E. Sutherland and Masamichi Takesaki. The intrinsic invariant of an approximately finite dimensional factor and the cocycle conjugacy of discrete amenable group actions. Electronic Research Announcements, 1995, 1: 43-47. |
[18] |
Russell Ricks. The unique measure of maximal entropy for a compact rank one locally CAT(0) space. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 507-523. doi: 10.3934/dcds.2020266 |
[19] |
Qiying Hu, Chen Xu, Wuyi Yue. A unified model for state feedback of discrete event systems I: framework and maximal permissive state feedback. Journal of Industrial and Management Optimization, 2008, 4 (1) : 107-123. doi: 10.3934/jimo.2008.4.107 |
[20] |
Byungik Kahng, Miguel Mendes. The characterization of maximal invariant sets of non-linear discrete-time control dynamical systems. Conference Publications, 2013, 2013 (special) : 393-406. doi: 10.3934/proc.2013.2013.393 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]