Singular Spectrum of Lebesgue Measure Zero¶for One-Dimensional Quasicrystals
Abstract: The spectrum of one-dimensional discrete Schrödinger operators associated to strictly ergodic dynamical systems is shown to coincide with the set of zeros of the Lyapunov exponent if and only if the Lyapunov exponent exists uniformly. This is used to obtain the Cantor spectrum of zero Lebe...
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Gespeichert in:
| Autor*in: |
Lenz, Daniel [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2002 |
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| Schlagwörter: |
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| Anmerkung: |
© Springer-Verlag Berlin Heidelberg 2002 |
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| Übergeordnetes Werk: |
Enthalten in: Communications in mathematical physics - Springer-Verlag, 1965, 227(2002), 1 vom: Mai, Seite 119-130 |
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| Übergeordnetes Werk: |
volume:227 ; year:2002 ; number:1 ; month:05 ; pages:119-130 |
| Links: |
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| DOI / URN: |
10.1007/s002200200624 |
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| 520 | |a Abstract: The spectrum of one-dimensional discrete Schrödinger operators associated to strictly ergodic dynamical systems is shown to coincide with the set of zeros of the Lyapunov exponent if and only if the Lyapunov exponent exists uniformly. This is used to obtain the Cantor spectrum of zero Lebesgue measure for all aperiodic subshifts with uniform positive weights. This covers, in particular, all aperiodic subshifts arising from primitive substitutions including new examples such as e.g. the Rudin–Shapiro substitution. Our investigation is not based on trace maps. Instead it relies on an Oseledec type theorem due to A. Furman and a uniform ergodic theorem due to the author. | ||
| 650 | 4 | |a Dynamical System | |
| 650 | 4 | |a Lebesgue Measure | |
| 650 | 4 | |a Lyapunov Exponent | |
| 650 | 4 | |a Measure Zero | |
| 650 | 4 | |a Ergodic Theorem | |
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10.1007/s002200200624 doi (DE-627)OLC2038880255 (DE-He213)s002200200624-p DE-627 ger DE-627 rakwb eng 530 510 VZ Lenz, Daniel verfasserin aut Singular Spectrum of Lebesgue Measure Zero¶for One-Dimensional Quasicrystals 2002 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag Berlin Heidelberg 2002 Abstract: The spectrum of one-dimensional discrete Schrödinger operators associated to strictly ergodic dynamical systems is shown to coincide with the set of zeros of the Lyapunov exponent if and only if the Lyapunov exponent exists uniformly. This is used to obtain the Cantor spectrum of zero Lebesgue measure for all aperiodic subshifts with uniform positive weights. This covers, in particular, all aperiodic subshifts arising from primitive substitutions including new examples such as e.g. the Rudin–Shapiro substitution. Our investigation is not based on trace maps. Instead it relies on an Oseledec type theorem due to A. Furman and a uniform ergodic theorem due to the author. Dynamical System Lebesgue Measure Lyapunov Exponent Measure Zero Ergodic Theorem Enthalten in Communications in mathematical physics Springer-Verlag, 1965 227(2002), 1 vom: Mai, Seite 119-130 (DE-627)129555002 (DE-600)220443-5 (DE-576)015011755 0010-3616 nnns volume:227 year:2002 number:1 month:05 pages:119-130 https://doi.org/10.1007/s002200200624 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-PHY SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_22 GBV_ILN_30 GBV_ILN_40 GBV_ILN_70 GBV_ILN_120 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2010 GBV_ILN_2018 GBV_ILN_2088 GBV_ILN_2279 GBV_ILN_2409 GBV_ILN_4125 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4307 GBV_ILN_4310 GBV_ILN_4313 GBV_ILN_4318 GBV_ILN_4319 AR 227 2002 1 05 119-130 |
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10.1007/s002200200624 doi (DE-627)OLC2038880255 (DE-He213)s002200200624-p DE-627 ger DE-627 rakwb eng 530 510 VZ Lenz, Daniel verfasserin aut Singular Spectrum of Lebesgue Measure Zero¶for One-Dimensional Quasicrystals 2002 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag Berlin Heidelberg 2002 Abstract: The spectrum of one-dimensional discrete Schrödinger operators associated to strictly ergodic dynamical systems is shown to coincide with the set of zeros of the Lyapunov exponent if and only if the Lyapunov exponent exists uniformly. This is used to obtain the Cantor spectrum of zero Lebesgue measure for all aperiodic subshifts with uniform positive weights. This covers, in particular, all aperiodic subshifts arising from primitive substitutions including new examples such as e.g. the Rudin–Shapiro substitution. Our investigation is not based on trace maps. Instead it relies on an Oseledec type theorem due to A. Furman and a uniform ergodic theorem due to the author. Dynamical System Lebesgue Measure Lyapunov Exponent Measure Zero Ergodic Theorem Enthalten in Communications in mathematical physics Springer-Verlag, 1965 227(2002), 1 vom: Mai, Seite 119-130 (DE-627)129555002 (DE-600)220443-5 (DE-576)015011755 0010-3616 nnns volume:227 year:2002 number:1 month:05 pages:119-130 https://doi.org/10.1007/s002200200624 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-PHY SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_22 GBV_ILN_30 GBV_ILN_40 GBV_ILN_70 GBV_ILN_120 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2010 GBV_ILN_2018 GBV_ILN_2088 GBV_ILN_2279 GBV_ILN_2409 GBV_ILN_4125 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4307 GBV_ILN_4310 GBV_ILN_4313 GBV_ILN_4318 GBV_ILN_4319 AR 227 2002 1 05 119-130 |
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10.1007/s002200200624 doi (DE-627)OLC2038880255 (DE-He213)s002200200624-p DE-627 ger DE-627 rakwb eng 530 510 VZ Lenz, Daniel verfasserin aut Singular Spectrum of Lebesgue Measure Zero¶for One-Dimensional Quasicrystals 2002 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag Berlin Heidelberg 2002 Abstract: The spectrum of one-dimensional discrete Schrödinger operators associated to strictly ergodic dynamical systems is shown to coincide with the set of zeros of the Lyapunov exponent if and only if the Lyapunov exponent exists uniformly. This is used to obtain the Cantor spectrum of zero Lebesgue measure for all aperiodic subshifts with uniform positive weights. This covers, in particular, all aperiodic subshifts arising from primitive substitutions including new examples such as e.g. the Rudin–Shapiro substitution. Our investigation is not based on trace maps. Instead it relies on an Oseledec type theorem due to A. Furman and a uniform ergodic theorem due to the author. Dynamical System Lebesgue Measure Lyapunov Exponent Measure Zero Ergodic Theorem Enthalten in Communications in mathematical physics Springer-Verlag, 1965 227(2002), 1 vom: Mai, Seite 119-130 (DE-627)129555002 (DE-600)220443-5 (DE-576)015011755 0010-3616 nnns volume:227 year:2002 number:1 month:05 pages:119-130 https://doi.org/10.1007/s002200200624 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-PHY SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_22 GBV_ILN_30 GBV_ILN_40 GBV_ILN_70 GBV_ILN_120 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2010 GBV_ILN_2018 GBV_ILN_2088 GBV_ILN_2279 GBV_ILN_2409 GBV_ILN_4125 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4307 GBV_ILN_4310 GBV_ILN_4313 GBV_ILN_4318 GBV_ILN_4319 AR 227 2002 1 05 119-130 |
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10.1007/s002200200624 doi (DE-627)OLC2038880255 (DE-He213)s002200200624-p DE-627 ger DE-627 rakwb eng 530 510 VZ Lenz, Daniel verfasserin aut Singular Spectrum of Lebesgue Measure Zero¶for One-Dimensional Quasicrystals 2002 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag Berlin Heidelberg 2002 Abstract: The spectrum of one-dimensional discrete Schrödinger operators associated to strictly ergodic dynamical systems is shown to coincide with the set of zeros of the Lyapunov exponent if and only if the Lyapunov exponent exists uniformly. This is used to obtain the Cantor spectrum of zero Lebesgue measure for all aperiodic subshifts with uniform positive weights. This covers, in particular, all aperiodic subshifts arising from primitive substitutions including new examples such as e.g. the Rudin–Shapiro substitution. Our investigation is not based on trace maps. Instead it relies on an Oseledec type theorem due to A. Furman and a uniform ergodic theorem due to the author. Dynamical System Lebesgue Measure Lyapunov Exponent Measure Zero Ergodic Theorem Enthalten in Communications in mathematical physics Springer-Verlag, 1965 227(2002), 1 vom: Mai, Seite 119-130 (DE-627)129555002 (DE-600)220443-5 (DE-576)015011755 0010-3616 nnns volume:227 year:2002 number:1 month:05 pages:119-130 https://doi.org/10.1007/s002200200624 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-PHY SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_22 GBV_ILN_30 GBV_ILN_40 GBV_ILN_70 GBV_ILN_120 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2010 GBV_ILN_2018 GBV_ILN_2088 GBV_ILN_2279 GBV_ILN_2409 GBV_ILN_4125 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4307 GBV_ILN_4310 GBV_ILN_4313 GBV_ILN_4318 GBV_ILN_4319 AR 227 2002 1 05 119-130 |
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10.1007/s002200200624 doi (DE-627)OLC2038880255 (DE-He213)s002200200624-p DE-627 ger DE-627 rakwb eng 530 510 VZ Lenz, Daniel verfasserin aut Singular Spectrum of Lebesgue Measure Zero¶for One-Dimensional Quasicrystals 2002 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag Berlin Heidelberg 2002 Abstract: The spectrum of one-dimensional discrete Schrödinger operators associated to strictly ergodic dynamical systems is shown to coincide with the set of zeros of the Lyapunov exponent if and only if the Lyapunov exponent exists uniformly. This is used to obtain the Cantor spectrum of zero Lebesgue measure for all aperiodic subshifts with uniform positive weights. This covers, in particular, all aperiodic subshifts arising from primitive substitutions including new examples such as e.g. the Rudin–Shapiro substitution. Our investigation is not based on trace maps. Instead it relies on an Oseledec type theorem due to A. Furman and a uniform ergodic theorem due to the author. Dynamical System Lebesgue Measure Lyapunov Exponent Measure Zero Ergodic Theorem Enthalten in Communications in mathematical physics Springer-Verlag, 1965 227(2002), 1 vom: Mai, Seite 119-130 (DE-627)129555002 (DE-600)220443-5 (DE-576)015011755 0010-3616 nnns volume:227 year:2002 number:1 month:05 pages:119-130 https://doi.org/10.1007/s002200200624 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-PHY SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_22 GBV_ILN_30 GBV_ILN_40 GBV_ILN_70 GBV_ILN_120 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2010 GBV_ILN_2018 GBV_ILN_2088 GBV_ILN_2279 GBV_ILN_2409 GBV_ILN_4125 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4307 GBV_ILN_4310 GBV_ILN_4313 GBV_ILN_4318 GBV_ILN_4319 AR 227 2002 1 05 119-130 |
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10.1007/s002200200624 |
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singular spectrum of lebesgue measure zero¶for one-dimensional quasicrystals |
| title_auth |
Singular Spectrum of Lebesgue Measure Zero¶for One-Dimensional Quasicrystals |
| abstract |
Abstract: The spectrum of one-dimensional discrete Schrödinger operators associated to strictly ergodic dynamical systems is shown to coincide with the set of zeros of the Lyapunov exponent if and only if the Lyapunov exponent exists uniformly. This is used to obtain the Cantor spectrum of zero Lebesgue measure for all aperiodic subshifts with uniform positive weights. This covers, in particular, all aperiodic subshifts arising from primitive substitutions including new examples such as e.g. the Rudin–Shapiro substitution. Our investigation is not based on trace maps. Instead it relies on an Oseledec type theorem due to A. Furman and a uniform ergodic theorem due to the author. © Springer-Verlag Berlin Heidelberg 2002 |
| abstractGer |
Abstract: The spectrum of one-dimensional discrete Schrödinger operators associated to strictly ergodic dynamical systems is shown to coincide with the set of zeros of the Lyapunov exponent if and only if the Lyapunov exponent exists uniformly. This is used to obtain the Cantor spectrum of zero Lebesgue measure for all aperiodic subshifts with uniform positive weights. This covers, in particular, all aperiodic subshifts arising from primitive substitutions including new examples such as e.g. the Rudin–Shapiro substitution. Our investigation is not based on trace maps. Instead it relies on an Oseledec type theorem due to A. Furman and a uniform ergodic theorem due to the author. © Springer-Verlag Berlin Heidelberg 2002 |
| abstract_unstemmed |
Abstract: The spectrum of one-dimensional discrete Schrödinger operators associated to strictly ergodic dynamical systems is shown to coincide with the set of zeros of the Lyapunov exponent if and only if the Lyapunov exponent exists uniformly. This is used to obtain the Cantor spectrum of zero Lebesgue measure for all aperiodic subshifts with uniform positive weights. This covers, in particular, all aperiodic subshifts arising from primitive substitutions including new examples such as e.g. the Rudin–Shapiro substitution. Our investigation is not based on trace maps. Instead it relies on an Oseledec type theorem due to A. Furman and a uniform ergodic theorem due to the author. © Springer-Verlag Berlin Heidelberg 2002 |
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Singular Spectrum of Lebesgue Measure Zero¶for One-Dimensional Quasicrystals |
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