Almost Everything About the Unitary Almost Mathieu Operator Article Swipe
Christopher Cedzich
,
Jake Fillman
,
Darren C. Ong
·
YOU?
·
· 2021
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2112.03216
YOU?
·
· 2021
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2112.03216
We introduce a unitary almost-Mathieu operator, which is obtained from a two-dimensional quantum walk in a uniform magnetic field. We exhibit a version of Aubry--André duality for this model, which partitions the parameter space into three regions: a supercritical region and a subcritical region that are dual to one another, and a critical regime that is self-dual. In each parameter region, we characterize the cocycle dynamics of the transfer matrix cocycle generated by the associated generalized eigenvalue equation. In particular, we show that supercritical, critical, and subcritical behavior all occur in this model. Using Avila's global theory of one-frequency cocycles, we exactly compute the Lyapunov exponent on the spectrum in terms of the given parameters. We also characterize the spectral type for each value of the coupling constant, almost every frequency, and almost every phase. Namely, we show that for almost every frequency and every phase the spectral type is purely absolutely continuous in the subcritical region, pure point in the supercritical region, and purely singular continuous in the critical region. In some parameter regions, we refine the almost-sure results. In the critical case for instance, we show that the spectrum is a Cantor set of zero Lebesgue measure for arbitrary irrational frequency and that the spectrum is purely singular continuous for all but countably many phases.
Related Topics
Concepts
Mathematics
Lyapunov exponent
Lebesgue measure
Spectrum (functional analysis)
Continuous spectrum
Operator (biology)
Absolute continuity
Eigenvalues and eigenvectors
Critical point (mathematics)
Pure mathematics
Mathematical analysis
Mathematical physics
Quantum mechanics
Physics
Nonlinear system
Lebesgue integration
Gene
Chemistry
Repressor
Transcription factor
Biochemistry
Metadata
- Type
- preprint
- Language
- en
- Landing Page
- http://arxiv.org/abs/2112.03216
- https://arxiv.org/pdf/2112.03216
- OA Status
- green
- Related Works
- 10
- OpenAlex ID
- https://openalex.org/W4200635626
All OpenAlex metadata
Raw OpenAlex JSON
- OpenAlex ID
-
https://openalex.org/W4200635626Canonical identifier for this work in OpenAlex
- DOI
-
https://doi.org/10.48550/arxiv.2112.03216Digital Object Identifier
- Title
-
Almost Everything About the Unitary Almost Mathieu OperatorWork title
- Type
-
preprintOpenAlex work type
- Language
-
enPrimary language
- Publication year
-
2021Year of publication
- Publication date
-
2021-12-06Full publication date if available
- Authors
-
Christopher Cedzich, Jake Fillman, Darren C. OngList of authors in order
- Landing page
-
https://arxiv.org/abs/2112.03216Publisher landing page
- PDF URL
-
https://arxiv.org/pdf/2112.03216Direct link to full text PDF
- Open access
-
YesWhether a free full text is available
- OA status
-
greenOpen access status per OpenAlex
- OA URL
-
https://arxiv.org/pdf/2112.03216Direct OA link when available
- Concepts
-
Mathematics, Lyapunov exponent, Lebesgue measure, Spectrum (functional analysis), Continuous spectrum, Operator (biology), Absolute continuity, Eigenvalues and eigenvectors, Critical point (mathematics), Pure mathematics, Mathematical analysis, Mathematical physics, Quantum mechanics, Physics, Nonlinear system, Lebesgue integration, Gene, Chemistry, Repressor, Transcription factor, BiochemistryTop concepts (fields/topics) attached by OpenAlex
- Cited by
-
0Total citation count in OpenAlex
- Related works (count)
-
10Other works algorithmically related by OpenAlex
Full payload
| id | https://openalex.org/W4200635626 |
|---|---|
| doi | https://doi.org/10.48550/arxiv.2112.03216 |
| ids.doi | https://doi.org/10.48550/arxiv.2112.03216 |
| ids.openalex | https://openalex.org/W4200635626 |
| fwci | 0.0 |
| type | preprint |
| title | Almost Everything About the Unitary Almost Mathieu Operator |
| biblio.issue | |
| biblio.volume | |
| biblio.last_page | |
| biblio.first_page | |
| topics[0].id | https://openalex.org/T10382 |
| topics[0].field.id | https://openalex.org/fields/31 |
| topics[0].field.display_name | Physics and Astronomy |
| topics[0].score | 0.9998000264167786 |
| topics[0].domain.id | https://openalex.org/domains/3 |
| topics[0].domain.display_name | Physical Sciences |
| topics[0].subfield.id | https://openalex.org/subfields/3107 |
| topics[0].subfield.display_name | Atomic and Molecular Physics, and Optics |
| topics[0].display_name | Quantum and electron transport phenomena |
| topics[1].id | https://openalex.org/T10682 |
| topics[1].field.id | https://openalex.org/fields/17 |
| topics[1].field.display_name | Computer Science |
| topics[1].score | 0.9997000098228455 |
| topics[1].domain.id | https://openalex.org/domains/3 |
| topics[1].domain.display_name | Physical Sciences |
| topics[1].subfield.id | https://openalex.org/subfields/1702 |
| topics[1].subfield.display_name | Artificial Intelligence |
| topics[1].display_name | Quantum Computing Algorithms and Architecture |
| topics[2].id | https://openalex.org/T10020 |
| topics[2].field.id | https://openalex.org/fields/17 |
| topics[2].field.display_name | Computer Science |
| topics[2].score | 0.9990000128746033 |
| topics[2].domain.id | https://openalex.org/domains/3 |
| topics[2].domain.display_name | Physical Sciences |
| topics[2].subfield.id | https://openalex.org/subfields/1702 |
| topics[2].subfield.display_name | Artificial Intelligence |
| topics[2].display_name | Quantum Information and Cryptography |
| is_xpac | False |
| apc_list | |
| apc_paid | |
| concepts[0].id | https://openalex.org/C33923547 |
| concepts[0].level | 0 |
| concepts[0].score | 0.6104369163513184 |
| concepts[0].wikidata | https://www.wikidata.org/wiki/Q395 |
| concepts[0].display_name | Mathematics |
| concepts[1].id | https://openalex.org/C191544260 |
| concepts[1].level | 3 |
| concepts[1].score | 0.588918924331665 |
| concepts[1].wikidata | https://www.wikidata.org/wiki/Q1238630 |
| concepts[1].display_name | Lyapunov exponent |
| concepts[2].id | https://openalex.org/C2777105136 |
| concepts[2].level | 3 |
| concepts[2].score | 0.5869506597518921 |
| concepts[2].wikidata | https://www.wikidata.org/wiki/Q827230 |
| concepts[2].display_name | Lebesgue measure |
| concepts[3].id | https://openalex.org/C156778621 |
| concepts[3].level | 2 |
| concepts[3].score | 0.45523977279663086 |
| concepts[3].wikidata | https://www.wikidata.org/wiki/Q1365748 |
| concepts[3].display_name | Spectrum (functional analysis) |
| concepts[4].id | https://openalex.org/C80845027 |
| concepts[4].level | 2 |
| concepts[4].score | 0.45391541719436646 |
| concepts[4].wikidata | https://www.wikidata.org/wiki/Q15144062 |
| concepts[4].display_name | Continuous spectrum |
| concepts[5].id | https://openalex.org/C17020691 |
| concepts[5].level | 5 |
| concepts[5].score | 0.44470301270484924 |
| concepts[5].wikidata | https://www.wikidata.org/wiki/Q139677 |
| concepts[5].display_name | Operator (biology) |
| concepts[6].id | https://openalex.org/C118733216 |
| concepts[6].level | 2 |
| concepts[6].score | 0.4306415915489197 |
| concepts[6].wikidata | https://www.wikidata.org/wiki/Q332504 |
| concepts[6].display_name | Absolute continuity |
| concepts[7].id | https://openalex.org/C158693339 |
| concepts[7].level | 2 |
| concepts[7].score | 0.41348734498023987 |
| concepts[7].wikidata | https://www.wikidata.org/wiki/Q190524 |
| concepts[7].display_name | Eigenvalues and eigenvectors |
| concepts[8].id | https://openalex.org/C196298200 |
| concepts[8].level | 2 |
| concepts[8].score | 0.4127446413040161 |
| concepts[8].wikidata | https://www.wikidata.org/wiki/Q577705 |
| concepts[8].display_name | Critical point (mathematics) |
| concepts[9].id | https://openalex.org/C202444582 |
| concepts[9].level | 1 |
| concepts[9].score | 0.3691074252128601 |
| concepts[9].wikidata | https://www.wikidata.org/wiki/Q837863 |
| concepts[9].display_name | Pure mathematics |
| concepts[10].id | https://openalex.org/C134306372 |
| concepts[10].level | 1 |
| concepts[10].score | 0.35426944494247437 |
| concepts[10].wikidata | https://www.wikidata.org/wiki/Q7754 |
| concepts[10].display_name | Mathematical analysis |
| concepts[11].id | https://openalex.org/C37914503 |
| concepts[11].level | 1 |
| concepts[11].score | 0.33927029371261597 |
| concepts[11].wikidata | https://www.wikidata.org/wiki/Q156495 |
| concepts[11].display_name | Mathematical physics |
| concepts[12].id | https://openalex.org/C62520636 |
| concepts[12].level | 1 |
| concepts[12].score | 0.25898319482803345 |
| concepts[12].wikidata | https://www.wikidata.org/wiki/Q944 |
| concepts[12].display_name | Quantum mechanics |
| concepts[13].id | https://openalex.org/C121332964 |
| concepts[13].level | 0 |
| concepts[13].score | 0.23439842462539673 |
| concepts[13].wikidata | https://www.wikidata.org/wiki/Q413 |
| concepts[13].display_name | Physics |
| concepts[14].id | https://openalex.org/C158622935 |
| concepts[14].level | 2 |
| concepts[14].score | 0.1481560468673706 |
| concepts[14].wikidata | https://www.wikidata.org/wiki/Q660848 |
| concepts[14].display_name | Nonlinear system |
| concepts[15].id | https://openalex.org/C14158598 |
| concepts[15].level | 2 |
| concepts[15].score | 0.14668560028076172 |
| concepts[15].wikidata | https://www.wikidata.org/wiki/Q853831 |
| concepts[15].display_name | Lebesgue integration |
| concepts[16].id | https://openalex.org/C104317684 |
| concepts[16].level | 2 |
| concepts[16].score | 0.0 |
| concepts[16].wikidata | https://www.wikidata.org/wiki/Q7187 |
| concepts[16].display_name | Gene |
| concepts[17].id | https://openalex.org/C185592680 |
| concepts[17].level | 0 |
| concepts[17].score | 0.0 |
| concepts[17].wikidata | https://www.wikidata.org/wiki/Q2329 |
| concepts[17].display_name | Chemistry |
| concepts[18].id | https://openalex.org/C158448853 |
| concepts[18].level | 4 |
| concepts[18].score | 0.0 |
| concepts[18].wikidata | https://www.wikidata.org/wiki/Q425218 |
| concepts[18].display_name | Repressor |
| concepts[19].id | https://openalex.org/C86339819 |
| concepts[19].level | 3 |
| concepts[19].score | 0.0 |
| concepts[19].wikidata | https://www.wikidata.org/wiki/Q407384 |
| concepts[19].display_name | Transcription factor |
| concepts[20].id | https://openalex.org/C55493867 |
| concepts[20].level | 1 |
| concepts[20].score | 0.0 |
| concepts[20].wikidata | https://www.wikidata.org/wiki/Q7094 |
| concepts[20].display_name | Biochemistry |
| keywords[0].id | https://openalex.org/keywords/mathematics |
| keywords[0].score | 0.6104369163513184 |
| keywords[0].display_name | Mathematics |
| keywords[1].id | https://openalex.org/keywords/lyapunov-exponent |
| keywords[1].score | 0.588918924331665 |
| keywords[1].display_name | Lyapunov exponent |
| keywords[2].id | https://openalex.org/keywords/lebesgue-measure |
| keywords[2].score | 0.5869506597518921 |
| keywords[2].display_name | Lebesgue measure |
| keywords[3].id | https://openalex.org/keywords/spectrum |
| keywords[3].score | 0.45523977279663086 |
| keywords[3].display_name | Spectrum (functional analysis) |
| keywords[4].id | https://openalex.org/keywords/continuous-spectrum |
| keywords[4].score | 0.45391541719436646 |
| keywords[4].display_name | Continuous spectrum |
| keywords[5].id | https://openalex.org/keywords/operator |
| keywords[5].score | 0.44470301270484924 |
| keywords[5].display_name | Operator (biology) |
| keywords[6].id | https://openalex.org/keywords/absolute-continuity |
| keywords[6].score | 0.4306415915489197 |
| keywords[6].display_name | Absolute continuity |
| keywords[7].id | https://openalex.org/keywords/eigenvalues-and-eigenvectors |
| keywords[7].score | 0.41348734498023987 |
| keywords[7].display_name | Eigenvalues and eigenvectors |
| keywords[8].id | https://openalex.org/keywords/critical-point |
| keywords[8].score | 0.4127446413040161 |
| keywords[8].display_name | Critical point (mathematics) |
| keywords[9].id | https://openalex.org/keywords/pure-mathematics |
| keywords[9].score | 0.3691074252128601 |
| keywords[9].display_name | Pure mathematics |
| keywords[10].id | https://openalex.org/keywords/mathematical-analysis |
| keywords[10].score | 0.35426944494247437 |
| keywords[10].display_name | Mathematical analysis |
| keywords[11].id | https://openalex.org/keywords/mathematical-physics |
| keywords[11].score | 0.33927029371261597 |
| keywords[11].display_name | Mathematical physics |
| keywords[12].id | https://openalex.org/keywords/quantum-mechanics |
| keywords[12].score | 0.25898319482803345 |
| keywords[12].display_name | Quantum mechanics |
| keywords[13].id | https://openalex.org/keywords/physics |
| keywords[13].score | 0.23439842462539673 |
| keywords[13].display_name | Physics |
| keywords[14].id | https://openalex.org/keywords/nonlinear-system |
| keywords[14].score | 0.1481560468673706 |
| keywords[14].display_name | Nonlinear system |
| keywords[15].id | https://openalex.org/keywords/lebesgue-integration |
| keywords[15].score | 0.14668560028076172 |
| keywords[15].display_name | Lebesgue integration |
| language | en |
| locations[0].id | pmh:oai:arXiv.org:2112.03216 |
| locations[0].is_oa | True |
| locations[0].source.id | https://openalex.org/S4306400194 |
| locations[0].source.issn | |
| locations[0].source.type | repository |
| locations[0].source.is_oa | True |
| locations[0].source.issn_l | |
| locations[0].source.is_core | False |
| locations[0].source.is_in_doaj | False |
| locations[0].source.display_name | arXiv (Cornell University) |
| locations[0].source.host_organization | https://openalex.org/I205783295 |
| locations[0].source.host_organization_name | Cornell University |
| locations[0].source.host_organization_lineage | https://openalex.org/I205783295 |
| locations[0].license | |
| locations[0].pdf_url | https://arxiv.org/pdf/2112.03216 |
| locations[0].version | submittedVersion |
| locations[0].raw_type | text |
| locations[0].license_id | |
| locations[0].is_accepted | False |
| locations[0].is_published | False |
| locations[0].raw_source_name | |
| locations[0].landing_page_url | http://arxiv.org/abs/2112.03216 |
| locations[1].id | doi:10.48550/arxiv.2112.03216 |
| locations[1].is_oa | True |
| locations[1].source.id | https://openalex.org/S4306400194 |
| locations[1].source.issn | |
| locations[1].source.type | repository |
| locations[1].source.is_oa | True |
| locations[1].source.issn_l | |
| locations[1].source.is_core | False |
| locations[1].source.is_in_doaj | False |
| locations[1].source.display_name | arXiv (Cornell University) |
| locations[1].source.host_organization | https://openalex.org/I205783295 |
| locations[1].source.host_organization_name | Cornell University |
| locations[1].source.host_organization_lineage | https://openalex.org/I205783295 |
| locations[1].license | |
| locations[1].pdf_url | |
| locations[1].version | |
| locations[1].raw_type | article-journal |
| locations[1].license_id | |
| locations[1].is_accepted | False |
| locations[1].is_published | |
| locations[1].raw_source_name | |
| locations[1].landing_page_url | https://doi.org/10.48550/arxiv.2112.03216 |
| indexed_in | arxiv, datacite |
| authorships[0].author.id | https://openalex.org/A5078157931 |
| authorships[0].author.orcid | |
| authorships[0].author.display_name | Christopher Cedzich |
| authorships[0].author_position | first |
| authorships[0].raw_author_name | Cedzich, Christopher |
| authorships[0].is_corresponding | False |
| authorships[1].author.id | https://openalex.org/A5012762299 |
| authorships[1].author.orcid | https://orcid.org/0000-0003-4716-710X |
| authorships[1].author.display_name | Jake Fillman |
| authorships[1].author_position | middle |
| authorships[1].raw_author_name | Fillman, Jake |
| authorships[1].is_corresponding | False |
| authorships[2].author.id | https://openalex.org/A5077311285 |
| authorships[2].author.orcid | https://orcid.org/0000-0003-4942-4376 |
| authorships[2].author.display_name | Darren C. Ong |
| authorships[2].author_position | last |
| authorships[2].raw_author_name | Ong, Darren C. |
| authorships[2].is_corresponding | False |
| has_content.pdf | False |
| has_content.grobid_xml | False |
| is_paratext | False |
| open_access.is_oa | True |
| open_access.oa_url | https://arxiv.org/pdf/2112.03216 |
| open_access.oa_status | green |
| open_access.any_repository_has_fulltext | False |
| created_date | 2025-10-10T00:00:00 |
| display_name | Almost Everything About the Unitary Almost Mathieu Operator |
| has_fulltext | False |
| is_retracted | False |
| updated_date | 2025-11-06T06:51:31.235846 |
| primary_topic.id | https://openalex.org/T10382 |
| primary_topic.field.id | https://openalex.org/fields/31 |
| primary_topic.field.display_name | Physics and Astronomy |
| primary_topic.score | 0.9998000264167786 |
| primary_topic.domain.id | https://openalex.org/domains/3 |
| primary_topic.domain.display_name | Physical Sciences |
| primary_topic.subfield.id | https://openalex.org/subfields/3107 |
| primary_topic.subfield.display_name | Atomic and Molecular Physics, and Optics |
| primary_topic.display_name | Quantum and electron transport phenomena |
| related_works | https://openalex.org/W1995524689, https://openalex.org/W2018078545, https://openalex.org/W2067225253, https://openalex.org/W4320560303, https://openalex.org/W2597963304, https://openalex.org/W2052524090, https://openalex.org/W3136108429, https://openalex.org/W3166422169, https://openalex.org/W4300603661, https://openalex.org/W2952246015 |
| cited_by_count | 0 |
| locations_count | 2 |
| best_oa_location.id | pmh:oai:arXiv.org:2112.03216 |
| best_oa_location.is_oa | True |
| best_oa_location.source.id | https://openalex.org/S4306400194 |
| best_oa_location.source.issn | |
| best_oa_location.source.type | repository |
| best_oa_location.source.is_oa | True |
| best_oa_location.source.issn_l | |
| best_oa_location.source.is_core | False |
| best_oa_location.source.is_in_doaj | False |
| best_oa_location.source.display_name | arXiv (Cornell University) |
| best_oa_location.source.host_organization | https://openalex.org/I205783295 |
| best_oa_location.source.host_organization_name | Cornell University |
| best_oa_location.source.host_organization_lineage | https://openalex.org/I205783295 |
| best_oa_location.license | |
| best_oa_location.pdf_url | https://arxiv.org/pdf/2112.03216 |
| best_oa_location.version | submittedVersion |
| best_oa_location.raw_type | text |
| best_oa_location.license_id | |
| best_oa_location.is_accepted | False |
| best_oa_location.is_published | False |
| best_oa_location.raw_source_name | |
| best_oa_location.landing_page_url | http://arxiv.org/abs/2112.03216 |
| primary_location.id | pmh:oai:arXiv.org:2112.03216 |
| primary_location.is_oa | True |
| primary_location.source.id | https://openalex.org/S4306400194 |
| primary_location.source.issn | |
| primary_location.source.type | repository |
| primary_location.source.is_oa | True |
| primary_location.source.issn_l | |
| primary_location.source.is_core | False |
| primary_location.source.is_in_doaj | False |
| primary_location.source.display_name | arXiv (Cornell University) |
| primary_location.source.host_organization | https://openalex.org/I205783295 |
| primary_location.source.host_organization_name | Cornell University |
| primary_location.source.host_organization_lineage | https://openalex.org/I205783295 |
| primary_location.license | |
| primary_location.pdf_url | https://arxiv.org/pdf/2112.03216 |
| primary_location.version | submittedVersion |
| primary_location.raw_type | text |
| primary_location.license_id | |
| primary_location.is_accepted | False |
| primary_location.is_published | False |
| primary_location.raw_source_name | |
| primary_location.landing_page_url | http://arxiv.org/abs/2112.03216 |
| publication_date | 2021-12-06 |
| publication_year | 2021 |
| referenced_works_count | 0 |
| abstract_inverted_index.a | 2, 10, 15, 21, 37, 41, 51, 192 |
| abstract_inverted_index.In | 57, 78, 171, 180 |
| abstract_inverted_index.We | 0, 19, 115 |
| abstract_inverted_index.by | 72 |
| abstract_inverted_index.in | 14, 90, 109, 153, 159, 167 |
| abstract_inverted_index.is | 7, 55, 149, 191, 207 |
| abstract_inverted_index.of | 23, 66, 97, 111, 124, 195 |
| abstract_inverted_index.on | 106 |
| abstract_inverted_index.to | 47 |
| abstract_inverted_index.we | 61, 80, 100, 136, 175, 186 |
| abstract_inverted_index.all | 88, 212 |
| abstract_inverted_index.and | 40, 50, 85, 131, 143, 163, 203 |
| abstract_inverted_index.are | 45 |
| abstract_inverted_index.but | 213 |
| abstract_inverted_index.for | 26, 121, 139, 184, 199, 211 |
| abstract_inverted_index.one | 48 |
| abstract_inverted_index.set | 194 |
| abstract_inverted_index.the | 31, 63, 67, 73, 103, 107, 112, 118, 125, 146, 154, 160, 168, 177, 181, 189, 205 |
| abstract_inverted_index.also | 116 |
| abstract_inverted_index.case | 183 |
| abstract_inverted_index.dual | 46 |
| abstract_inverted_index.each | 58, 122 |
| abstract_inverted_index.from | 9 |
| abstract_inverted_index.into | 34 |
| abstract_inverted_index.many | 215 |
| abstract_inverted_index.pure | 157 |
| abstract_inverted_index.show | 81, 137, 187 |
| abstract_inverted_index.some | 172 |
| abstract_inverted_index.that | 44, 54, 82, 138, 188, 204 |
| abstract_inverted_index.this | 27, 91 |
| abstract_inverted_index.type | 120, 148 |
| abstract_inverted_index.walk | 13 |
| abstract_inverted_index.zero | 196 |
| abstract_inverted_index.Using | 93 |
| abstract_inverted_index.every | 129, 133, 141, 144 |
| abstract_inverted_index.given | 113 |
| abstract_inverted_index.occur | 89 |
| abstract_inverted_index.phase | 145 |
| abstract_inverted_index.point | 158 |
| abstract_inverted_index.space | 33 |
| abstract_inverted_index.terms | 110 |
| abstract_inverted_index.three | 35 |
| abstract_inverted_index.value | 123 |
| abstract_inverted_index.which | 6, 29 |
| abstract_inverted_index.Cantor | 193 |
| abstract_inverted_index.almost | 128, 132, 140 |
| abstract_inverted_index.field. | 18 |
| abstract_inverted_index.global | 95 |
| abstract_inverted_index.matrix | 69 |
| abstract_inverted_index.model, | 28 |
| abstract_inverted_index.model. | 92 |
| abstract_inverted_index.phase. | 134 |
| abstract_inverted_index.purely | 150, 164, 208 |
| abstract_inverted_index.refine | 176 |
| abstract_inverted_index.regime | 53 |
| abstract_inverted_index.region | 39, 43 |
| abstract_inverted_index.theory | 96 |
| abstract_inverted_index.Avila's | 94 |
| abstract_inverted_index.Namely, | 135 |
| abstract_inverted_index.cocycle | 64, 70 |
| abstract_inverted_index.compute | 102 |
| abstract_inverted_index.duality | 25 |
| abstract_inverted_index.exactly | 101 |
| abstract_inverted_index.exhibit | 20 |
| abstract_inverted_index.measure | 198 |
| abstract_inverted_index.phases. | 216 |
| abstract_inverted_index.quantum | 12 |
| abstract_inverted_index.region, | 60, 156, 162 |
| abstract_inverted_index.region. | 170 |
| abstract_inverted_index.uniform | 16 |
| abstract_inverted_index.unitary | 3 |
| abstract_inverted_index.version | 22 |
| abstract_inverted_index.Lebesgue | 197 |
| abstract_inverted_index.Lyapunov | 104 |
| abstract_inverted_index.another, | 49 |
| abstract_inverted_index.behavior | 87 |
| abstract_inverted_index.coupling | 126 |
| abstract_inverted_index.critical | 52, 169, 182 |
| abstract_inverted_index.dynamics | 65 |
| abstract_inverted_index.exponent | 105 |
| abstract_inverted_index.magnetic | 17 |
| abstract_inverted_index.obtained | 8 |
| abstract_inverted_index.regions, | 174 |
| abstract_inverted_index.regions: | 36 |
| abstract_inverted_index.results. | 179 |
| abstract_inverted_index.singular | 165, 209 |
| abstract_inverted_index.spectral | 119, 147 |
| abstract_inverted_index.spectrum | 108, 190, 206 |
| abstract_inverted_index.transfer | 68 |
| abstract_inverted_index.arbitrary | 200 |
| abstract_inverted_index.cocycles, | 99 |
| abstract_inverted_index.constant, | 127 |
| abstract_inverted_index.countably | 214 |
| abstract_inverted_index.critical, | 84 |
| abstract_inverted_index.equation. | 77 |
| abstract_inverted_index.frequency | 142, 202 |
| abstract_inverted_index.generated | 71 |
| abstract_inverted_index.instance, | 185 |
| abstract_inverted_index.introduce | 1 |
| abstract_inverted_index.operator, | 5 |
| abstract_inverted_index.parameter | 32, 59, 173 |
| abstract_inverted_index.absolutely | 151 |
| abstract_inverted_index.associated | 74 |
| abstract_inverted_index.continuous | 152, 166, 210 |
| abstract_inverted_index.eigenvalue | 76 |
| abstract_inverted_index.frequency, | 130 |
| abstract_inverted_index.irrational | 201 |
| abstract_inverted_index.partitions | 30 |
| abstract_inverted_index.self-dual. | 56 |
| abstract_inverted_index.almost-sure | 178 |
| abstract_inverted_index.generalized | 75 |
| abstract_inverted_index.parameters. | 114 |
| abstract_inverted_index.particular, | 79 |
| abstract_inverted_index.subcritical | 42, 86, 155 |
| abstract_inverted_index.characterize | 62, 117 |
| abstract_inverted_index.Aubry--André | 24 |
| abstract_inverted_index.one-frequency | 98 |
| abstract_inverted_index.supercritical | 38, 161 |
| abstract_inverted_index.almost-Mathieu | 4 |
| abstract_inverted_index.supercritical, | 83 |
| abstract_inverted_index.two-dimensional | 11 |
| cited_by_percentile_year | |
| countries_distinct_count | 0 |
| institutions_distinct_count | 3 |
| citation_normalized_percentile.value | 0.17640537 |
| citation_normalized_percentile.is_in_top_1_percent | False |
| citation_normalized_percentile.is_in_top_10_percent | False |