메뉴 건너뛰기




Volumn 82, Issue 1, 2007, Pages 61-77

Pure point diffraction implies zero entropy for delone sets with uniform cluster frequencies

Author keywords

Delone sets; Entropy; Model sets; Substitutions

Indexed keywords


EID: 36549002808     PISSN: 03779017     EISSN: 15730530     Source Type: Journal    
DOI: 10.1007/s11005-007-0186-7     Document Type: Article
Times cited : (40)

References (55)
  • 1
    • 0000331178 scopus 로고
    • The third law of thermodynamics and the degeneracy of the ground state for lattice systems
    • Aizenman M. and Lieb E.H. (1981). The third law of thermodynamics and the degeneracy of the ground state for lattice systems. J. Stat. Phys. 24: 279-297
    • (1981) J. Stat. Phys. , vol.24 , pp. 279-297
    • Aizenman, M.1    Lieb, E.H.2
  • 2
    • 36549062086 scopus 로고
    • Beyond Quasicrystals
    • Axel, F., Gratias, D. (eds.) Les Ulis, and Springer, Berlin
    • Axel, F., Gratias, D. (eds.): Beyond Quasicrystals, Les Editions de Physique, Les Ulis, and Springer, Berlin (1995)
    • (1995) Les Editions de Physique
  • 3
    • 0013431406 scopus 로고    scopus 로고
    • A guide to mathematical quasicrystals
    • Suck, J.-B., Schreiber, M., Häussler, P. (eds.) Quasicrystals Springer, Berlin arXiv:math-ph/9901014
    • Baake, M.: A guide to mathematical quasicrystals. In: Suck, J.-B., Schreiber, M., Häussler, P. (eds.) Quasicrystals. Springer Series in Materials Science, vol. 55, pp. 17-48 Springer, Berlin (2002); arXiv:math-ph/9901014
    • (2002) Springer Series in Materials Science , vol.55 , pp. 17-48
    • Baake, M.1
  • 4
    • 0031557837 scopus 로고    scopus 로고
    • The torus parametrization of quasiperiodic LI classes
    • mp_arc/02-168
    • Baake M., Hermisson J. and Pleasants P.A.B. (1997). The torus parametrization of quasiperiodic LI classes. J. Phys. A Math. Gen. 30: 3029-3056, mp_arc/02-168
    • (1997) J. Phys. A Math. Gen. , vol.30 , pp. 3029-3056
    • Baake, M.1    Hermisson, J.2    Pleasants, P.A.B.3
  • 5
    • 0034339817 scopus 로고    scopus 로고
    • Diffraction of random tilings: Some rigorous results
    • arXiv:math-ph/9904005
    • Baake M. and Höffe M. (2000). Diffraction of random tilings: some rigorous results. J. Stat. Phys. 99: 219-261, arXiv:math-ph/9904005
    • (2000) J. Stat. Phys. , vol.99 , pp. 219-261
    • Baake, M.1    Höffe, M.2
  • 6
    • 10844288676 scopus 로고    scopus 로고
    • Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra, Ergod
    • arXiv:math.DS/0302231
    • Baake M. and Lenz D. (2004). Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra, Ergod. Th. Dyn. Syst. 24: 1867-1893, arXiv:math.DS/0302231
    • (2004) Th. Dyn. Syst. , vol.24 , pp. 1867-1893
    • Baake, M.1    Lenz, D.2
  • 7
    • 33947244989 scopus 로고    scopus 로고
    • Charactererization of model sets by dynamical systems
    • arXiv:math.DS/0511648
    • Baake M., Lenz D. and Moody R.V. (2007). Charactererization of model sets by dynamical systems. Ergod. Th. Dyn. Syst. 27: 341-382, arXiv:math.DS/0511648
    • (2007) Ergod. Th. Dyn. Syst. , vol.27 , pp. 341-382
    • Baake, M.1    Lenz, D.2    Moody, R.V.3
  • 8
    • 0000498157 scopus 로고    scopus 로고
    • Directions in Mathematical Quasicrystals
    • Baake, M., Moody, R.V (eds.) AMS, Providence
    • Baake, M., Moody, R.V (eds.): Directions in Mathematical Quasicrystals. CRM Monograph Series, vol. 13. AMS, Providence (2000)
    • (2000) CRM Monograph Series , vol.13
  • 9
    • 0011973787 scopus 로고    scopus 로고
    • Diffraction from visible lattice points and k-th power free integers
    • arXiv:math.MG/9906132
    • Baake M., Moody R.V. and Pleasants P.A.B. (2000). Diffraction from visible lattice points and k-th power free integers. Discret. Math. 221: 3-42, arXiv:math.MG/9906132
    • (2000) Discret. Math. , vol.221 , pp. 3-42
    • Baake, M.1    Moody, R.V.2    Pleasants, P.A.B.3
  • 10
    • 0040194908 scopus 로고
    • Strictly ergodic Toeplitz flows with positive entropies and trivial centralizers
    • Bulatek W. and Kwiatowski J. (1992). Strictly ergodic Toeplitz flows with positive entropies and trivial centralizers. Stud. Math. 103: 133-142
    • (1992) Stud. Math. , vol.103 , pp. 133-142
    • Bulatek, W.1    Kwiatowski, J.2
  • 13
    • 0039010995 scopus 로고
    • A criterion for Toeplitz flows to be topologically isomorphic and applications
    • Downarowicz T., Kwiatowski J. and Lacroix Y. (1995). A criterion for Toeplitz flows to be topologically isomorphic and applications. Coll. Math. 68: 219-228
    • (1995) Coll. Math. , vol.68 , pp. 219-228
    • Downarowicz, T.1    Kwiatowski, J.2    Lacroix, Y.3
  • 14
    • 0034413138 scopus 로고    scopus 로고
    • Linearly recurrent subshifts have a finite number of non-periodic subshift factors
    • Durand F. (2000). Linearly recurrent subshifts have a finite number of non-periodic subshift factors. Ergod. Th. Dyn. Syst. 20: 1061-1078
    • (2000) Ergod. Th. Dyn. Syst. , vol.20 , pp. 1061-1078
    • Durand, F.1
  • 15
    • 36549078729 scopus 로고
    • 2 -action on a compact metric space
    • 2 -action on a compact metric space. Am. J. Math. 99: 77-106
    • (1977) Am. J. Math. , vol.99 , pp. 77-106
    • Elsanousi, S.A.1
  • 16
    • 33646474152 scopus 로고    scopus 로고
    • Algebraic topology for minimal Cantor sets
    • Gambaudo J.-M. and Martens M. (2006). Algebraic topology for minimal Cantor sets. Ann. H. Poincaré 7: 423-446
    • (2006) Ann. H. Poincaré , vol.7 , pp. 423-446
    • Gambaudo, J.-M.1    Martens, M.2
  • 17
    • 17144385117 scopus 로고    scopus 로고
    • Quasicrystals and almost periodicity
    • arXiv:math-ph/0212012
    • Gouéré J.-B. (2005). Quasicrystals and almost periodicity. Commun. Math. Phys. 255: 655-681, arXiv:math-ph/0212012
    • (2005) Commun. Math. Phys. , vol.255 , pp. 655-681
    • Gouéré, J.-B.1
  • 18
    • 84968503501 scopus 로고
    • On the entropy of uniquely ergodic transformations
    • Hahn F. and Katznelson Y. (1967). On the entropy of uniquely ergodic transformations. Trans. AMS 126: 335-360
    • (1967) Trans. AMS , vol.126 , pp. 335-360
    • Hahn, F.1    Katznelson, Y.2
  • 20
    • 0031212140 scopus 로고    scopus 로고
    • A guide to the symmetry structure of quasiperiodic tiling classes
    • mp_arc/02-180
    • Hermisson J., Richard C. and Baake M. (1997). A guide to the symmetry structure of quasiperiodic tiling classes. J. Phys. I (Fr.) 7: 1003-1018, mp_arc/02-180
    • (1997) J. Phys. i (Fr.) , vol.7 , pp. 1003-1018
    • Hermisson, J.1    Richard, C.2    Baake, M.3
  • 21
    • 21844524319 scopus 로고
    • On diffraction by aperiodic structures
    • Hof A. (1995). On diffraction by aperiodic structures. Commun. Math. Phys. 169: 25-43
    • (1995) Commun. Math. Phys. , vol.169 , pp. 25-43
    • Hof, A.1
  • 22
    • 0033809458 scopus 로고    scopus 로고
    • Surprises in diffuse scattering
    • arXiv:math-ph/0004022
    • Höffe M. and Baake M. (2000). Surprises in diffuse scattering. Z. Kristallogr. 215: 441-444, arXiv:math-ph/0004022
    • (2000) Z. Kristallogr. , vol.215 , pp. 441-444
    • Höffe, M.1    Baake, M.2
  • 23
    • 0001163746 scopus 로고
    • Dimer statistics and phase transitions
    • Kasteleyn P.W. (1963). Dimer statistics and phase transitions. J. Math. Phys. 4: 287-293
    • (1963) J. Math. Phys. , vol.4 , pp. 287-293
    • Kasteleyn, P.W.1
  • 25
    • 0041411813 scopus 로고    scopus 로고
    • Repetitive Delone sets and quasicrystals
    • arXiv:math.DS/9909033
    • Lagarias J.C. and Pleasants P.A.B. (2003). Repetitive Delone sets and quasicrystals. Ergod. Th. Dyn. Syst. 23: 831-867, arXiv:math.DS/9909033
    • (2003) Ergod. Th. Dyn. Syst. , vol.23 , pp. 831-867
    • Lagarias, J.C.1    Pleasants, P.A.B.2
  • 26
    • 0036898223 scopus 로고    scopus 로고
    • Local complexity of Delone sets and crystallinity
    • arXiv:math.MG/0105088
    • Lagarias J.C. and Pleasants P.A.B. (2002). Local complexity of Delone sets and crystallinity. Canad. Math. Bull. 45: 634-652, arXiv:math.MG/0105088
    • (2002) Canad. Math. Bull. , vol.45 , pp. 634-652
    • Lagarias, J.C.1    Pleasants, P.A.B.2
  • 27
    • 0036435987 scopus 로고    scopus 로고
    • Pure point dynamical and diffraction spectra
    • mp_arc/02-39
    • Lee J.-Y., Moody R.V. and Solomyak B. (2002). Pure point dynamical and diffraction spectra. Ann. H. Poincaré 3: 1003-1018, mp_arc/02-39
    • (2002) Ann. H. Poincaré , vol.3 , pp. 1003-1018
    • Lee, J.-Y.1    Moody, R.V.2    Solomyak, B.3
  • 28
    • 1642345030 scopus 로고    scopus 로고
    • Aperiodic linearly repetitive sets are densely repetitive
    • arXiv:math.DS/9909033
    • Lenz D. (2004). Aperiodic linearly repetitive sets are densely repetitive. Discrete Comput. Geom. 31: 323-326, arXiv:math.DS/9909033
    • (2004) Discrete Comput. Geom. , vol.31 , pp. 323-326
    • Lenz, D.1
  • 29
    • 3543012017 scopus 로고    scopus 로고
    • Algebras of random operators associated to Delone dynamical systems
    • arXiv:math-ph/0210031
    • Lenz D. and Stollmann P. (2003). Algebras of random operators associated to Delone dynamical systems, Math. Phys. Anal. Geom. 6: 269-290, arXiv:math-ph/0210031
    • (2003) Math. Phys. Anal. Geom. , vol.6 , pp. 269-290
    • Lenz, D.1    Stollmann, P.2
  • 31
    • 0000635276 scopus 로고
    • Mahler measure and entropy for commuting automorphisms of compact groups
    • Lind D., Schmidt K. and Ward T. (1990). Mahler measure and entropy for commuting automorphisms of compact groups. Invent. Math. 101: 593-629
    • (1990) Invent. Math. , vol.101 , pp. 593-629
    • Lind, D.1    Schmidt, K.2    Ward, T.3
  • 32
    • 0002421044 scopus 로고    scopus 로고
    • The Mathematics of Long-Range Aperiodic Order
    • Moody, R.V (ed.) Kluwer, Dordrecht
    • Moody, R.V (ed.): The Mathematics of Long-Range Aperiodic Order. NATO ASI Series C 489. Kluwer, Dordrecht (1997)
    • (1997) NATO ASI Series C , vol.489
  • 33
    • 0013330815 scopus 로고    scopus 로고
    • Model sets: A survey
    • Axel, F., Dénoyer, F., Gazeau, J.P. (eds.) EDP Sciences, Les Ulis, and Springer, Berlin arXiv:math.MG/0002020
    • Moody, R.V.: Model sets: A survey, In: Axel, F., Dénoyer, F., Gazeau, J.P. (eds.) From Quasicrystals to More Complex Systems, pp. 145-166, EDP Sciences, Les Ulis, and Springer, Berlin (2000); arXiv:math.MG/0002020
    • (2000) From Quasicrystals to More Complex Systems , pp. 145-166
    • Moody, R.V.1
  • 34
    • 51249182088 scopus 로고
    • Almost automorphic symbolic minimal sets without unique ergodicity
    • Paul M.E. and Markley N.G. (1979). Almost automorphic symbolic minimal sets without unique ergodicity. Isr. J. Math. 34: 259-272
    • (1979) Isr. J. Math. , vol.34 , pp. 259-272
    • Paul, M.E.1    Markley, N.G.2
  • 36
    • 0003831421 scopus 로고
    • Cambridge University Press Cambridge
    • Petersen K. (1983). Ergodic Theory. Cambridge University Press, Cambridge
    • (1983) Ergodic Theory
    • Petersen, K.1
  • 39
    • 0038930773 scopus 로고    scopus 로고
    • Random tilings: Concepts and examples
    • arXiv:cond-mat/9712267
    • Richard C., Höffe M., Hermisson J. and Baake M. (1998). Random tilings: concepts and examples. J. Phys. A Math. Gen. 31: 6385-6408, arXiv:cond-mat/9712267
    • (1998) J. Phys. A Math. Gen. , vol.31 , pp. 6385-6408
    • Richard, C.1    Höffe, M.2    Hermisson, J.3    Baake, M.4
  • 40
    • 0038929066 scopus 로고    scopus 로고
    • An alternative view on quasicrystalline random tilings
    • arXiv:cond-mat/9907262
    • Richard C. (1999). An alternative view on quasicrystalline random tilings. J. Phys. A Math. Gen. 32: 8823-8829, arXiv:cond-mat/9907262
    • (1999) J. Phys. A Math. Gen. , vol.32 , pp. 8823-8829
    • Richard, C.1
  • 41
    • 0142073347 scopus 로고    scopus 로고
    • Dense Dirac combs in Euclidean space with pure point diffraction
    • arXiv:math-ph/0302049
    • Richard C. (2003). Dense Dirac combs in Euclidean space with pure point diffraction. J. Math. Phys. 44: 4436-4449, arXiv:math-ph/0302049
    • (2003) J. Math. Phys. , vol.44 , pp. 4436-4449
    • Richard, C.1
  • 42
    • 33845734511 scopus 로고    scopus 로고
    • d
    • Williams, S.G. (ed.) Symbolic Dynamics and its Applications AMS, Providence, RI (2004)
    • d. In: Williams, S.G. (ed.) Symbolic Dynamics and its Applications, Proc. Sympos. Appl. Math., vol. 60, AMS, Providence, RI (2004), pp. 81-119 (2004)
    • (2004) Proc. Sympos. Appl. Math. , vol.60 , pp. 81-119
    • Robinson Jr., E.A.1
  • 44
    • 34548851064 scopus 로고    scopus 로고
    • 2 Cambridge University Press Cambridge
    • Ruelle D. (2004). Thermodynamic Formalism, 2nd edn. Cambridge University Press, Cambridge
    • (2004) Thermodynamic Formalism
    • Ruelle, D.1
  • 45
    • 33751141500 scopus 로고
    • Metallic phase with long-range orientational order and no translational symmetry
    • Shechtman D., Blech I., Gratias D. and Cahn J.W. (1984). Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53: 1951-1953
    • (1984) Phys. Rev. Lett. , vol.53 , pp. 1951-1953
    • Shechtman, D.1    Blech, I.2    Gratias, D.3    Cahn, J.W.4
  • 49
    • 0031506167 scopus 로고    scopus 로고
    • Dynamics of self-similar tilings
    • Erratum: Ergod. Th. Dynam. Syst. 19, 1685 (1999)
    • Solomyak, B.: Dynamics of self-similar tilings. Ergod. Th. Dyn. Syst. 17, 695-738 (1997) Erratum: Ergod. Th. Dynam. Syst. 19, 1685 (1999)
    • (1997) Ergod. Th. Dyn. Syst. , vol.17 , pp. 695-738
    • Solomyak, B.1
  • 50
    • 0040350273 scopus 로고    scopus 로고
    • Nonperiodicity implies unique composition for self-similar translationally finite tilings
    • Solomyak B. (1998). Nonperiodicity implies unique composition for self-similar translationally finite tilings. Discrete Comput. Geom. 20: 265-279
    • (1998) Discrete Comput. Geom. , vol.20 , pp. 265-279
    • Solomyak, B.1
  • 51
    • 33745825293 scopus 로고    scopus 로고
    • On coincidence of entropies for two classes of dynamical systems
    • Solomyak R. (1998). On coincidence of entropies for two classes of dynamical systems. Ergod. Th. Dyn. Syst. 18: 731-738
    • (1998) Ergod. Th. Dyn. Syst. , vol.18 , pp. 731-738
    • Solomyak, R.1
  • 53
    • 34249928982 scopus 로고
    • translation in Math. Notes 49 (1991), 305-311
    • (1991) Math. Notes , vol.49 , pp. 305-311
  • 55
    • 36049059211 scopus 로고
    • Remarks on the modified potassium dihydrogen phosphate model of a ferroelectric
    • Wu F.Y. (1968). Remarks on the modified potassium dihydrogen phosphate model of a ferroelectric. Phys. Rev. 168: 539-543
    • (1968) Phys. Rev. , vol.168 , pp. 539-543
    • Wu, F.Y.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.