Causal Holography of Traversing Flows

Abstract

We study smooth traversing vector fields v on compact manifolds X with boundary. A traversing v admits a Lyapunov function f:X→R such that df(v)>0. We show that the trajectory spaces T(v) of traversally generic v-flows are Whitney stratified spaces, and thus admit triangulations amenable to their natural stratifications. Despite being spaces with singularities, T(v) retain some residual smooth structure of X. Let F(v) denote the oriented 1-dimensional foliation on X, produced by a traversing v-flow. With the help of a boundary generic v, we divide the boundary ∂X of X into two complementary compact manifolds, ∂+X(v) and ∂−X(v). Then, for a traversing v, we introduce the causality map Cv:∂+X(v)→∂−X(v). Our main result claims that, for boundary generic traversing vector fields v, the causality map Cv allows for a reconstruction of the pair (X,F(v)), up to a homeomorphism Φ:X→X such that Φ|∂X=id∂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 rigid. We call these results “holographic” since the (n+1)-dimensional X and the un-parameterized dynamics of the v-flow are captured by a single map Cv between two n-dimensional screens, ∂+X(v) and ∂−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.