# 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…
9 Citations
• G. Katz
• Mathematics
Qualitative Theory of Dynamical Systems
• 2020
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
Let $X$ be a smooth compact manifold and $v$ a vector field on $X$ which admits a smooth function $f: X \to \mathbf R$ such that $df(v)>0$. Let $\partial X$ be the boundary of $X$. We denote by
For a traversing vector field $v$ on a compact $(n+1)$-manifold $X$ with boundary, we use closed $v$-invariant differential $n$-forms $\Theta$ to define measures $\mu_\Theta$ on the boundary
• G. Katz
• Mathematics
Inverse Problems & Imaging
• 2019
For a given smooth compact manifold $M$, we introduce an open class $\mathcal G(M)$ of Riemannian metrics, which we call \emph{metrics of the gradient type}. For such metrics $g$, the geodesic flow
We combine Gromov's amenable localization technique with the Poincar\'{e} duality to study the traversally generic vector flows on smooth compact manifolds $X$ with boundary. Such flows generate
Let $M$ be a compact connected smooth Riemannian $n$-manifold with boundary. We combine Gromov's amenable localization technique with the Poincar\'{e} duality to study the traversally generic
• G. Katz
• Materials Science
Qualitative Theory of Dynamical Systems
• 2021
Let M be a compact smooth Riemannian n-manifold with boundary. We combine Gromov’s amenable localization technique with the Poincaré duality to study the traversally generic geodesic flows on SM, the
• Mathematics
Chaos
• 2022
Given a closed k-dimensional submanifold K, encapsulated in a compact domain M ⊂ E, k ≤ n - 2, we consider the problem of determining the intrinsic geometry of the obstacle K (such as volume,
• G. Katz
• Mathematics, Computer Science
• 2015
This paper uses the relative simplicity of 2-dimensional worlds to popularize the approach to the Morse theory on smooth manifolds with boundary, and takes advantage of the boundary effects to take the central stage.

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