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.

中文翻译：

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基于球的折纸定理和用于流动的全息图

本文描述了一种机制，通过该机制，在具有边界的平滑连接\（（（n + 1）\））维流形X上的遍历通用流v产生紧凑的n维CW-复杂\（{\ mathcal {T}}（ v）\），它与X等价，并且X嵌入\（{\ mathcal {T}}（v）\ times \ mathbb R \）中。的CW -配合物\（\ mathcal T（V）\）捕获关于在光滑结构中的一些残留信息X（如稳定切丛X）。此外，\（{\ mathcal {T}}（v）\）是从简单折纸图\（O：D ^ n \ rightarrow {\ mathcal {T}}（v）\）中获得的，其源空间是一个球\（D ^ n \ subset \ partial X \）。O的纤维最多具有基数\（（n + 1）\）。映射O的知识，以及对v的Lyapunov函数\（f：X \ rightarrow \ mathbb R \）的\（D ^ n \）的限制，使得可以重建该对\ n的拓扑类型（（X，{\ mathcal {F}}（v））\）是\（{\ mathcal {F}}（v）\）是由v生成的1叶。这一事实促使在标题中使用“全息”一词。在全息原理的定性表述中，对于给定紧致流形X上的大量ODE ，解的适当阶段的边值问题的解决方案在拓扑上是刚性的。