The Ball-Based Origami Theorem and a Glimpse of Holography for Traversing Flows
Abstract This paper describes a mechanism by which a traversally generic flow v on a smooth connected $$(n+1)$$(n+1)-dimensional manifold X with boundary produces a compact n-dimensional CW-complex $${\mathcal {T}}(v)$$T(v), which is homotopy equivalent to X and such that X embeds in...
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| Format: | Article |
| Language: | English |
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Springer International Publishing
2021
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| Online Access: | https://hdl.handle.net/1721.1/131464 |
| _version_ | 1826214727967047680 |
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| author | Katz, Gabriel |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Katz, Gabriel |
| author_sort | Katz, Gabriel |
| collection | MIT |
| description | Abstract
This paper describes a mechanism by which a traversally generic flow v on a smooth connected $$(n+1)$$(n+1)-dimensional manifold X with boundary produces a compact n-dimensional CW-complex $${\mathcal {T}}(v)$$T(v), which is homotopy equivalent to X and such that X embeds in $${\mathcal {T}}(v)\times \mathbb R$$T(v)×R. The CW-complex $$\mathcal T(v)$$T(v) captures some residual information about the smooth structure on X (such as the stable tangent bundle of X). Moreover, $${\mathcal {T}}(v)$$T(v) is obtained from a simplicial origami map$$O: D^n \rightarrow {\mathcal {T}}(v)$$O:Dn→T(v), whose source space is a ball $$D^n \subset \partial X$$Dn⊂∂X. The fibers of O have the cardinality $$(n+1)$$(n+1) at most. The knowledge of the map O, together with the restriction to $$D^n$$Dn of a Lyapunov function $$f: X \rightarrow \mathbb R$$f:X→R for v, make it possible to reconstruct the topological type of the pair $$(X, {\mathcal {F}}(v))$$(X,F(v)), were $${\mathcal {F}}(v)$$F(v) is the 1-foliation, generated by v. This fact motivates the use of the word “holography” in the title. In a qualitative formulation of the holography principle, for a massive class of ODE’s on a given compact manifold X, the solutions of the appropriately staged boundary value problems are topologically rigid. |
| first_indexed | 2024-09-23T16:10:13Z |
| format | Article |
| id | mit-1721.1/131464 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T16:10:13Z |
| publishDate | 2021 |
| publisher | Springer International Publishing |
| record_format | dspace |
| spelling | mit-1721.1/1314642023-03-15T20:47:22Z The Ball-Based Origami Theorem and a Glimpse of Holography for Traversing Flows Katz, Gabriel Massachusetts Institute of Technology. Department of Mathematics Abstract This paper describes a mechanism by which a traversally generic flow v on a smooth connected $$(n+1)$$(n+1)-dimensional manifold X with boundary produces a compact n-dimensional CW-complex $${\mathcal {T}}(v)$$T(v), which is homotopy equivalent to X and such that X embeds in $${\mathcal {T}}(v)\times \mathbb R$$T(v)×R. The CW-complex $$\mathcal T(v)$$T(v) captures some residual information about the smooth structure on X (such as the stable tangent bundle of X). Moreover, $${\mathcal {T}}(v)$$T(v) is obtained from a simplicial origami map$$O: D^n \rightarrow {\mathcal {T}}(v)$$O:Dn→T(v), whose source space is a ball $$D^n \subset \partial X$$Dn⊂∂X. The fibers of O have the cardinality $$(n+1)$$(n+1) at most. The knowledge of the map O, together with the restriction to $$D^n$$Dn of a Lyapunov function $$f: X \rightarrow \mathbb R$$f:X→R for v, make it possible to reconstruct the topological type of the pair $$(X, {\mathcal {F}}(v))$$(X,F(v)), were $${\mathcal {F}}(v)$$F(v) is the 1-foliation, generated by v. This fact motivates the use of the word “holography” in the title. In a qualitative formulation of the holography principle, for a massive class of ODE’s on a given compact manifold X, the solutions of the appropriately staged boundary value problems are topologically rigid. 2021-09-20T17:17:11Z 2021-09-20T17:17:11Z 2020-02-07 2020-09-24T21:16:12Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/131464 Qualitative Theory of Dynamical Systems. 2020 Feb 07;19(1):41 en https://doi.org/10.1007/s12346-020-00364-7 Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. Springer Nature Switzerland AG application/pdf Springer International Publishing Springer International Publishing |
| spellingShingle | Katz, Gabriel The Ball-Based Origami Theorem and a Glimpse of Holography for Traversing Flows |
| title | The Ball-Based Origami Theorem and a Glimpse of Holography for Traversing Flows |
| title_full | The Ball-Based Origami Theorem and a Glimpse of Holography for Traversing Flows |
| title_fullStr | The Ball-Based Origami Theorem and a Glimpse of Holography for Traversing Flows |
| title_full_unstemmed | The Ball-Based Origami Theorem and a Glimpse of Holography for Traversing Flows |
| title_short | The Ball-Based Origami Theorem and a Glimpse of Holography for Traversing Flows |
| title_sort | ball based origami theorem and a glimpse of holography for traversing flows |
| url | https://hdl.handle.net/1721.1/131464 |
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