Causal Holography of Traversing Flows
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Causal Holography of Traversing Flows Gabriel Katz1
Received: 4 January 2017 / Revised: 23 September 2020 / Accepted: 14 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
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 d f (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 : X → X such that |∂ 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. Keywords Traversing vector flows · Manifolds with boundary · Causality maps · Boundary data · Holography
1 Introduction This paper is an extension of the sequence [12–15], which studies non-vanishing gradient-like flows on smooth compact manifolds with boundary. Our approach emphasizes the interactions of the flow trajectories with the boundary. Let X be a compact connected smooth (n + 1)-dimensional manifold with boundary. A smooth vector field v on X is called traversing if each v-trajectory is homeomorphic either
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Gabriel Katz [email protected] Massachusetts Institute of Technology, Department of Mathematics, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
123
Journal of Dynamics and Differential Equations
to a closed interval, or to a singleton. An equivalent definition of a traversing v is based on the existence of a Lyapunov function f : X → R such that d f (v) > 0 in X . In particular, the gradient flow of a Bott-Morse function f is traversing in the compliment to any open neighborhood of its critical set. The paper consists of five sections, including the Introduction. In Sect. 2, we introduce various classes of vector fields on manifolds with boundary and summarize their properties, needed for the rest of the paper. They include traversing, boundary generic, and traversally generic vector fields. In Sect. 3, we employ the semi-local algebraic mode
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