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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 \((n+1)\)-dimensional manifold X with boundary produces a compact n-dimensional CW-complex \({\mathcal {T}}(v)\), which is homotopy equivalent to X and such that X embeds in \({\mathcal {T}}(v)\times \mathbb R\). The CW-complex \(\mathcal T(v)\) captures some residual information about the smooth structure on X (such as the stable tangent bundle of X). Moreover, \({\mathcal {T}}(v)\) is obtained from a simplicial origami map\(O: D^n \rightarrow {\mathcal {T}}(v)\), whose source space is a ball \(D^n \subset \partial X\). The fibers of O have the cardinality \((n+1)\) at most. The knowledge of the map O, together with the restriction to \(D^n\) of a Lyapunov function \(f: X \rightarrow \mathbb R\) for v, make it possible to reconstruct the topological type of the pair \((X, {\mathcal {F}}(v))\), were \({\mathcal {F}}(v)\) 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.

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Notes

  1. i.e., compact 1-dimensional CW-complexes.

  2. In fact, we may assume that \(D^n \subset \partial X\).

  3. \(C^\infty ({\mathcal {T}}(v))\) is just a subalgebra of \(C^\infty (X)\), not an ideal.

  4. that is induced by the origami map \(\Gamma : D^n \rightarrow {\mathcal {T}}(v)\)

  5. see Definition 3.1.

  6. By Theorem 4.1, such vector field exists.

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Acknowledgements

The author is grateful to the referee for advising to make the original text more self-contained and for encouraging the author to explain better the motivation behind the results.

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Correspondence to Gabriel Katz.

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Katz, G. The Ball-Based Origami Theorem and a Glimpse of Holography for Traversing Flows. Qual. Theory Dyn. Syst. 19, 41 (2020). https://doi.org/10.1007/s12346-020-00364-7

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Keywords

  • Manifolds with boundary
  • Traversing flows
  • Trajectory spaces
  • Boundary value problems
  • Holography