Causal Holography of Traversing Flows

TL;DR

This paper demonstrates that for certain boundary-generic traversing flows on manifolds, the entire topology and flow dynamics can be reconstructed from a causality map between boundary components, revealing a holographic principle in flow topology.

Contribution

It introduces the concept of holography for traversing flows, showing that the causality map on the boundary uniquely determines the manifold and flow structure.

Findings

Trajectory spaces are Whitney stratified, allowing triangulation.

Causality map enables reconstruction of the manifold and flow.

Results apply to a broad class of boundary value problems.

Abstract

We study smooth {\sf traversing} vector fields $v$ on compact manifolds $X$ with boundary. A traversing $v$ admits a Lyapunov function $f: X \to \Bbb R$ such that $df(v) > 0$. We show that the trajectory spaces $\mathcal T(v)$ of {\sf traversally generic} $v$-flows are {\sf Whitney stratified spaces}, and thus admit triangulations amenable to their natural stratifications. Despite being spaces with singularities, $\mathcal T(v)$ retain some residual smooth structure of $X$. Let $\mathcal F(v)$ denote the oriented $1$-dimensional foliation on $X$, produced by a traversing $v$-flow. With the help of a {\sf boundary generic} $v$, we divide the boundary $\partial X$ of $X$ into two complementary compact manifolds, $\partial^+X(v)$ and $\partial^-X(v)$. Then, for a traversing $v$, we introduce the {\sf causality map} $C_v: \partial^+X(v) \to \partial^-X(v)$. Our main result claims that, for boundary generic traversing vector fields $v$, the causality map $C_v$ is allows for a reconstruction of the pair $(X, \mathcal F(v))$, up to a homeomorphism $\Phi: X \to X$ such that $\Phi|_{\partial X} = id_{\partial X}$. In other words, for a massive class of ODEs, we show that the topology of their solutions, satisfying a given boundary value problem, is {\sf rigid}. We call these results ``{\sf holographic}" since the $(n+1)$-dimensional $X$ and the un-parameterized dynamics of the $v$-flow are captured by a single map $C_v$ between two $n$-dimensional screens, $\partial^+X(v)$ and $\partial^-X(v)$. This holography of traversing flows has numerous applications to the dynamics of general flows. Some of them are described in the paper. Others, are just outlined.