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

Abstract

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.