The Schwarzian Octahedron Recurrence (dSKP Equation) II: Geometric Systems.
Saved in:
| Title: | The Schwarzian Octahedron Recurrence (dSKP Equation) II: Geometric Systems. |
|---|---|
| Authors: | Affolter, Niklas Christoph1,2 (AUTHOR) affolter@posteo.net, de Tilière, Béatrice3 (AUTHOR) detiliere@ceremade.dauphine.fr, Melotti, Paul4 (AUTHOR) melotti@posteo.net |
| Source: | Discrete & Computational Geometry. Mar2025, Vol. 73 Issue 2, p370-436. 67p. |
| Subjects: | Projective geometry, Discrete geometry, Integrable systems, Systems theory, Differential geometry |
| Abstract: | We consider nine geometric systems: Miquel dynamics, P-nets, integrable cross-ratio maps, discrete holomorphic functions, orthogonal circle patterns, polygon recutting, circle intersection dynamics, (corrugated) pentagram maps and the short diagonal hyperplane map. Using a unified framework, for each system we prove an explicit expression for the solution as a function of the initial data; more precisely, we show that the solution is equal to the ratio of two partition functions of an oriented dimer model on an Aztec diamond whose face weights are constructed from the initial data. Then, we study the Devron property (Glick in J Geom Phys 87:161–189, 2015), which states the following: if the system starts from initial data that is singular for the backwards dynamics, this singularity is expected to reoccur after a finite number of steps of the forwards dynamics. Again, using a unified framework, we prove this Devron property for all of the above geometric systems, for different kinds of singular initial data. In doing so, we obtain new singularity results and also known ones (Glick in J Geom Phys 87:161–189, 2015; Yao in Glick's conjecture on the point of collapse of axis-aligned polygons under the pentagram maps (2014). Preprint arXiv:1410.7806). Our general method consists in proving that these nine geometric systems are all related to the Schwarzian octahedron recurrence (dSKP equation), and then to rely on the companion paper (Affolter et al. in Comb Theory 3(2), 2023), where we study this recurrence in general, prove explicit expressions and singularity results. [ABSTRACT FROM AUTHOR] |
| Copyright of Discrete & Computational Geometry is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
|
Full text is not displayed to guests.
Login for full access.
|
|
| FullText | Links: – Type: pdflink Text: Availability: 1 |
|---|---|
| Header | DbId: egs DbLabel: Engineering Source An: 183080251 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
| IllustrationInfo | |
| Items | – Name: Title Label: Title Group: Ti Data: The Schwarzian Octahedron Recurrence (dSKP Equation) II: Geometric Systems. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Affolter%2C+Niklas+Christoph%22">Affolter, Niklas Christoph</searchLink><relatesTo>1,2</relatesTo> (AUTHOR)<i> affolter@posteo.net</i><br /><searchLink fieldCode="AR" term="%22de+Tilière%2C+Béatrice%22">de Tilière, Béatrice</searchLink><relatesTo>3</relatesTo> (AUTHOR)<i> detiliere@ceremade.dauphine.fr</i><br /><searchLink fieldCode="AR" term="%22Melotti%2C+Paul%22">Melotti, Paul</searchLink><relatesTo>4</relatesTo> (AUTHOR)<i> melotti@posteo.net</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Discrete+%26+Computational+Geometry%22">Discrete & Computational Geometry</searchLink>. Mar2025, Vol. 73 Issue 2, p370-436. 67p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Projective+geometry%22">Projective geometry</searchLink><br /><searchLink fieldCode="DE" term="%22Discrete+geometry%22">Discrete geometry</searchLink><br /><searchLink fieldCode="DE" term="%22Integrable+systems%22">Integrable systems</searchLink><br /><searchLink fieldCode="DE" term="%22Systems+theory%22">Systems theory</searchLink><br /><searchLink fieldCode="DE" term="%22Differential+geometry%22">Differential geometry</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: We consider nine geometric systems: Miquel dynamics, P-nets, integrable cross-ratio maps, discrete holomorphic functions, orthogonal circle patterns, polygon recutting, circle intersection dynamics, (corrugated) pentagram maps and the short diagonal hyperplane map. Using a unified framework, for each system we prove an explicit expression for the solution as a function of the initial data; more precisely, we show that the solution is equal to the ratio of two partition functions of an oriented dimer model on an Aztec diamond whose face weights are constructed from the initial data. Then, we study the Devron property (Glick in J Geom Phys 87:161–189, 2015), which states the following: if the system starts from initial data that is singular for the backwards dynamics, this singularity is expected to reoccur after a finite number of steps of the forwards dynamics. Again, using a unified framework, we prove this Devron property for all of the above geometric systems, for different kinds of singular initial data. In doing so, we obtain new singularity results and also known ones (Glick in J Geom Phys 87:161–189, 2015; Yao in Glick's conjecture on the point of collapse of axis-aligned polygons under the pentagram maps (2014). Preprint arXiv:1410.7806). Our general method consists in proving that these nine geometric systems are all related to the Schwarzian octahedron recurrence (dSKP equation), and then to rely on the companion paper (Affolter et al. in Comb Theory 3(2), 2023), where we study this recurrence in general, prove explicit expressions and singularity results. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Discrete & Computational Geometry is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
| PLink | https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=egs&AN=183080251 |
| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1007/s00454-024-00640-2 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 67 StartPage: 370 Subjects: – SubjectFull: Projective geometry Type: general – SubjectFull: Discrete geometry Type: general – SubjectFull: Integrable systems Type: general – SubjectFull: Systems theory Type: general – SubjectFull: Differential geometry Type: general Titles: – TitleFull: The Schwarzian Octahedron Recurrence (dSKP Equation) II: Geometric Systems. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Affolter, Niklas Christoph – PersonEntity: Name: NameFull: de Tilière, Béatrice – PersonEntity: Name: NameFull: Melotti, Paul IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 03 Text: Mar2025 Type: published Y: 2025 Identifiers: – Type: issn-print Value: 01795376 Numbering: – Type: volume Value: 73 – Type: issue Value: 2 Titles: – TitleFull: Discrete & Computational Geometry Type: main |
| ResultId | 1 |
