# 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} }

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

### The Ball-Based Origami Theorem and a Glimpse of Holography for Traversing Flows

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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…

### Algebras of smooth functions and holography of traversing flows

- Mathematics
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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…

### Holography of geodesic flows, harmonizing metrics, and billiards' dynamics

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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…

### Causal holography in application to the inverse scattering problems

- MathematicsInverse 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…

### Complexity of Shadows & Traversing Flows in Terms of the Simplicial Volume

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- 2015

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…

### Applying Gromov's Amenable Localization to Geodesic Flows

- Mathematics
- 2017

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…

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- Materials ScienceQualitative 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…

### Detecting intrinsic global geometry of an obstacle via layered scattering.

- MathematicsChaos
- 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,…

### Flows in Flatland: A Romance of Few Dimensions

- Mathematics, Computer Science
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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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