The Ball-Based Origami Theorem and a Glimpse of Holography for Traversing Flows

• Katz, Gabriel ^[1]
  1. [1] Massachusetts Institute of Technology

     Massachusetts Institute of Technology

     City of Cambridge, Estados Unidos

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 19, Nº 1, 2020
• Idioma: inglés
• DOI: 10.1007/s12346-020-00364-7
• Enlaces
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• Resumen

  • 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 T(v), which is homotopy equivalent to X and such that X embeds in T(v)×R. The CW-complex T(v) captures some residual information about the smooth structure on X (such as the stable tangent bundle of X). Moreover, T(v) is obtained from a simplicial origami mapO:Dn→T(v), whose source space is a ball Dn⊂∂X. The fibers of O have the cardinality (n+1) at most. The knowledge of the map O, together with the restriction to Dn of a Lyapunov function f:X→R for v, make it possible to reconstruct the topological type of the pair (X,F(v)), were 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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