• DOI:10.1007/s10884-020-09910-y
• Corpus ID: 119596346

Causal Holography of Traversing Flows

@article{Katz2014CausalHO,
  title={Causal Holography of Traversing Flows},
  author={Gabriel Katz},
  journal={Journal of Dynamics and Differential Equations},
  year={2014},
  volume={33},
  pages={235-274}
}

• G. Katz
• Published 2 September 2014
• Mathematics
• Journal of Dynamics and Differential Equations

We study smooth traversing vector fields v on compact manifolds X with boundary. A traversing v admits a Lyapunov function $$f: X \rightarrow \mathbb R$$ f : X → R such that $$df(v) > 0$$ d f ( v ) > 0 . We show that the trajectory spaces $${\mathcal {T}}(v)$$ 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, $${\mathcal {T}}(v)$$ T ( v ) retain some residual smooth structure of X . Let $${\mathcal {F}}(v)$$ 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 $$\partial X$$ ∂ X of X into two complementary compact manifolds, $$\partial ^+X(v)$$ ∂ + X ( v ) and $$\partial ^-X(v)$$ ∂ - X ( v ) . Then, for a traversing v , we introduce the causality map $$C_v: \partial ^+X(v) \rightarrow \partial ^-X(v)$$ C v : ∂ + X ( v ) → ∂ - X ( v ) . Our main result claims that, for boundary generic traversing vector fields v , the causality map $$C_v$$ C v allows for a reconstruction of the pair $$(X, {\mathcal {F}}(v))$$ ( X , F ( v ) ) , up to a homeomorphism $$\Phi : X \rightarrow X$$ Φ : X → X such that $$\Phi |_{\partial X} = id_{\partial X}$$ Φ | ∂ X = i d ∂ 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 $$(n+1)$$ ( n + 1 ) -dimensional X and the un-parameterized dynamics of the v -flow are captured by a single map $$C_v$$ C v between two n -dimensional screens, $$\partial ^+X(v)$$ ∂ + X ( v ) and $$\partial ^-X(v)$$ ∂ - 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.