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On the elliptic equation Δu+K up = 0 in $\mathbb{R}$n
Pure discrete spectrum in substitution tiling spaces
1. | Department of Mathematics, Montana State University, Bozeman, MT 59717, United States |
2. | Department of Mathematics, University of Zagreb, Bijenička 30, 10 000 Zagreb |
3. | Department of Mathematics, University of Texas, Austin, TX 78712, United States |
References:
[1] |
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. |
[2] |
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. |
[3] |
P. Arnoux and S. Ito, Pisot substitutions and Rauzy fractals, Bull. Belg. Math Soc., 8 (2001), 18-2007. |
[4] |
J. Auslander, "Minimal Flows and Their Extensions," North-Holland Mathematical Studies, 153, North-Holland, Amsterdam, New York, Oxford, and Tokyo, 1988. |
[5] |
M. Baake and R. V. Moody, Weighted Dirac combs with pure point diffraction, J. Reine Angew. Math., 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, J. Instit. Fourier, 56 (2006), 2213-2248.
doi: 10.5802/aif.2238. |
[7] |
M. Barge, H. Bruin, L. Jones and L. Sadun, Homological Pisot substitutions and exact regularity, To appear in Israel J. Math., preprint, arXiv:1001.2027. |
[8] |
M. Barge and J. Kellendonk, Proximality and pure point spectrum for tiling dynamical systems, preprint, arXiv:1108.4065. |
[9] |
M. Barge, J. Kellendonk and S. Schmeiding, Maximal equicontinuous factors and cohomology of tiling spaces, To appear in Fund. Math.,arXiv:1204.1432. |
[10] |
M. Barge and J. Kwapisz, Geometric theory of unimodular Pisot substitutions, Amer J. Math., 128 (2006), 1219-1282.
doi: 10.1353/ajm.2006.0037. |
[11] |
M. Barge and C. Olimb, Asymptotic structure in substitution tiling spaces, To appear in Ergodic Theory & Dynamical Systems, preprint, arXiv:1101.4902. |
[12] |
V. Berthé, T. Jolivet and A. Siegel, Substitutive Arnoux-Rauzy substitutions have pure discrete spectrum, preprint, arXiv:1108.5574. |
[13] |
V. Berthé and A. Siegel, Tilings associated with beta-numeration and substitutions, Integers: Electronic Journal of Combinatorial Number Theory, 5 (2005), A02.arXiv:1108.5574. |
[14] |
F. M. Dekking, The spectrum of dynamical systems arising from substitutions of constant length, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 41 (1978), 221-239. |
[15] |
S. Dworkin, Spectral theory and X-ray diffraction, J. Math. Phys., 34 (1993), 2965-2967.
doi: 10.1063/1.530108. |
[16] |
N. P. Fogg, "Substitutions in Dynamics, Arithmetics and Combinatorics," Lecture notes in mathematics, (eds. V. Berthé, S. Ferenczi, C. Mauduit and A. Siegel), Springer-Verlag, 2002. |
[17] |
D. Fretlöh and B. Sing, Computing modular coincidences for substitution tilings and point sets, Discrete Comput. Geom., 37 (2007), 381-407.
doi: 10.1007/s00454-006-1280-9. |
[18] |
S. Ito and H. Rao, Atomic surfaces, tiling and coincidence I. Irreducible case, Israel J. Math., 153 (2006), 129-156.
doi: 10.1007/BF02771781. |
[19] | |
[20] |
R. Kenyon and B. Solomyak, On the characterization of expansion maps for self-affine tilings, Discrete Comput. Geom., 43 (2010), 577-593. |
[21] |
J. Y. Lee, Substitution Delone multisets with pure point spectrum are inter-model sets, Journal of Geometry and Physics, 57 (2007), 2263-2285.
doi: 10.1016/j.geomphys.2007.07.003. |
[22] |
J. Y. Lee and R. Moody, Lattice substitution systems and model sets, Discrete Comput. Geom., 25 (2001), 173-201.
doi: 10.1007/s004540010083. |
[23] |
J. Y. Lee, R. Moody and B. Solomyak, Consequences of Pure Point Diffraction Spectra for Multiset Substitution Systems, Discrete Comp. Geom., 29 (2003), 525-560.
doi: 10.1007/s00454-003-0781-z. |
[24] |
J. Y. Lee and B. Solomyak, Pure point diffractive substitution Delone sets have the Meyer property, Discrete Comp. Geom., 34 (2008), 319-338.
doi: 10.1007/s00454-008-9054-1. |
[25] |
J. Y. Lee and B. Solomyak, Pisot family self-affine tilings, discrete spectrum, and the Meyer property, preprint, arXiv:1002.0039. |
[26] |
A. N. Livshits, Some examples of adic transformations and substitutions, Selecta Math. Sovietica, 11 (1992), 83-104. |
[27] |
P. Michel, Coincidence values and spectra of substitutions, Zeit. Wahr., 42 (1978), 205-227.
doi: 10.1007/BF00641410. |
[28] |
A. Siegel and J. Thuswaldner, Topological properties of Rauzy fractals, preprint. |
[29] |
V. F. Sirivent and B. Solomyak, Pure discrete spectrum for one-dimensional substitution systems of Pisot type, Canad. Math. Bull., 45 (2002), 697-710. Dedicated to Robert V. Moody.
doi: 10.4153/CMB-2002-062-3. |
[30] |
B. Solomyak, Nonperiodicity implies unique composition for self-similar translationally finite tilings, Discrete Comput. Geometry, 20 (1998), 265-279.
doi: 10.1007/PL00009386. |
[31] |
B. Solomyak, Eigenfunctions for substitution tiling systems, Advanced Studies in Pure Mathematics, 49 (2007), 433-454. |
[32] |
B. Solomyak, Dynamics of self-similar tilings, Ergodic Theory & Dynamical Systems, 17 (1997), 695-738.
doi: 10.1017/S0143385797084988. |
[33] |
W. A. Veech, The equicontinuous structure relation for minimal Abelian transformation groups, Amer. J. of Math. 90 (1968), 723-732. |
show all references
References:
[1] |
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. |
[2] |
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. |
[3] |
P. Arnoux and S. Ito, Pisot substitutions and Rauzy fractals, Bull. Belg. Math Soc., 8 (2001), 18-2007. |
[4] |
J. Auslander, "Minimal Flows and Their Extensions," North-Holland Mathematical Studies, 153, North-Holland, Amsterdam, New York, Oxford, and Tokyo, 1988. |
[5] |
M. Baake and R. V. Moody, Weighted Dirac combs with pure point diffraction, J. Reine Angew. Math., 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, J. Instit. Fourier, 56 (2006), 2213-2248.
doi: 10.5802/aif.2238. |
[7] |
M. Barge, H. Bruin, L. Jones and L. Sadun, Homological Pisot substitutions and exact regularity, To appear in Israel J. Math., preprint, arXiv:1001.2027. |
[8] |
M. Barge and J. Kellendonk, Proximality and pure point spectrum for tiling dynamical systems, preprint, arXiv:1108.4065. |
[9] |
M. Barge, J. Kellendonk and S. Schmeiding, Maximal equicontinuous factors and cohomology of tiling spaces, To appear in Fund. Math.,arXiv:1204.1432. |
[10] |
M. Barge and J. Kwapisz, Geometric theory of unimodular Pisot substitutions, Amer J. Math., 128 (2006), 1219-1282.
doi: 10.1353/ajm.2006.0037. |
[11] |
M. Barge and C. Olimb, Asymptotic structure in substitution tiling spaces, To appear in Ergodic Theory & Dynamical Systems, preprint, arXiv:1101.4902. |
[12] |
V. Berthé, T. Jolivet and A. Siegel, Substitutive Arnoux-Rauzy substitutions have pure discrete spectrum, preprint, arXiv:1108.5574. |
[13] |
V. Berthé and A. Siegel, Tilings associated with beta-numeration and substitutions, Integers: Electronic Journal of Combinatorial Number Theory, 5 (2005), A02.arXiv:1108.5574. |
[14] |
F. M. Dekking, The spectrum of dynamical systems arising from substitutions of constant length, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 41 (1978), 221-239. |
[15] |
S. Dworkin, Spectral theory and X-ray diffraction, J. Math. Phys., 34 (1993), 2965-2967.
doi: 10.1063/1.530108. |
[16] |
N. P. Fogg, "Substitutions in Dynamics, Arithmetics and Combinatorics," Lecture notes in mathematics, (eds. V. Berthé, S. Ferenczi, C. Mauduit and A. Siegel), Springer-Verlag, 2002. |
[17] |
D. Fretlöh and B. Sing, Computing modular coincidences for substitution tilings and point sets, Discrete Comput. Geom., 37 (2007), 381-407.
doi: 10.1007/s00454-006-1280-9. |
[18] |
S. Ito and H. Rao, Atomic surfaces, tiling and coincidence I. Irreducible case, Israel J. Math., 153 (2006), 129-156.
doi: 10.1007/BF02771781. |
[19] | |
[20] |
R. Kenyon and B. Solomyak, On the characterization of expansion maps for self-affine tilings, Discrete Comput. Geom., 43 (2010), 577-593. |
[21] |
J. Y. Lee, Substitution Delone multisets with pure point spectrum are inter-model sets, Journal of Geometry and Physics, 57 (2007), 2263-2285.
doi: 10.1016/j.geomphys.2007.07.003. |
[22] |
J. Y. Lee and R. Moody, Lattice substitution systems and model sets, Discrete Comput. Geom., 25 (2001), 173-201.
doi: 10.1007/s004540010083. |
[23] |
J. Y. Lee, R. Moody and B. Solomyak, Consequences of Pure Point Diffraction Spectra for Multiset Substitution Systems, Discrete Comp. Geom., 29 (2003), 525-560.
doi: 10.1007/s00454-003-0781-z. |
[24] |
J. Y. Lee and B. Solomyak, Pure point diffractive substitution Delone sets have the Meyer property, Discrete Comp. Geom., 34 (2008), 319-338.
doi: 10.1007/s00454-008-9054-1. |
[25] |
J. Y. Lee and B. Solomyak, Pisot family self-affine tilings, discrete spectrum, and the Meyer property, preprint, arXiv:1002.0039. |
[26] |
A. N. Livshits, Some examples of adic transformations and substitutions, Selecta Math. Sovietica, 11 (1992), 83-104. |
[27] |
P. Michel, Coincidence values and spectra of substitutions, Zeit. Wahr., 42 (1978), 205-227.
doi: 10.1007/BF00641410. |
[28] |
A. Siegel and J. Thuswaldner, Topological properties of Rauzy fractals, preprint. |
[29] |
V. F. Sirivent and B. Solomyak, Pure discrete spectrum for one-dimensional substitution systems of Pisot type, Canad. Math. Bull., 45 (2002), 697-710. Dedicated to Robert V. Moody.
doi: 10.4153/CMB-2002-062-3. |
[30] |
B. Solomyak, Nonperiodicity implies unique composition for self-similar translationally finite tilings, Discrete Comput. Geometry, 20 (1998), 265-279.
doi: 10.1007/PL00009386. |
[31] |
B. Solomyak, Eigenfunctions for substitution tiling systems, Advanced Studies in Pure Mathematics, 49 (2007), 433-454. |
[32] |
B. Solomyak, Dynamics of self-similar tilings, Ergodic Theory & Dynamical Systems, 17 (1997), 695-738.
doi: 10.1017/S0143385797084988. |
[33] |
W. A. Veech, The equicontinuous structure relation for minimal Abelian transformation groups, Amer. J. of Math. 90 (1968), 723-732. |
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