Causal Holography of Traversing Flows

  title={Causal Holography of Traversing Flows},
  author={Gabriel Katz},
  journal={Journal of Dynamics and Differential Equations},
  • 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… 

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

  • 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

Algebras of smooth functions and holography of traversing flows

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

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

  • 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

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

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

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

Gromov’s Amenable Localization and Geodesic Flows

  • 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

Detecting intrinsic global geometry of an obstacle via layered scattering.

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

  • 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.



Convexity of Morse Stratifications and Spines of 3-Manifolds

The gradient fields $v$ of nonsingular functions $f$ on compact 3-folds $X$ with boundary are used to generate their spines $K(f, v)$. We study the transformations of $K(f, v)$ that are induced by

The Stratified Spaces of Real Polynomials & Trajectory Spaces of Traversing Flows

This paper is the third in a series that researches the Morse Theory, gradient flows, concavity and complexity on smooth compact manifolds with boundary. Employing the local analytic models from

Stratified Convexity & Concavity of Gradient Flows on Manifolds with Boundary

As has been observed by Morse [1], any generic vector field v on a compact smooth manifold X with boundary gives rise to a stratification of the boundary by compact submanifolds , where . Our main

Differentiable periodic maps

1. The bordism groups. This note presents an outline of the authors' efforts to apply Thorn's cobordism theory [ó] to the study of differentiable periodic maps. First, however, we shall outline our

Singularities of Mappings of Euclidean Spaces

We shall describe here some results and methods pertaining to the following general problem (details will appear elsewhere). Suppose a mapping f 0of an open set R in n-spaceE n into m-space E m is

On the triangulation of stratified sets and singular varieties

We show that every compact stratified set in the sense of Thom can be triangulated as a simplicial complex. The proof uses that author's description of a stratified set as the geometric realisation

The Kervaire invariant of framed manifolds and its generalization

In 1960, Kervaire [11] introduced an invariant for almost framed (4k + 2)manifolds, (k # 0, 1, 3), and proved that it was zero for framed 10-manifolds, which was a key step in his construction of a

Minimal entropy and Mostow's rigidity theorems

Let (Y, g) be a compact connected n-dimensional Riemannian manifold and let () be its universal cover endowed with the pulled-back metric. If y ∈ , we define where B(y, R) denotes the ball of radius

Manifolds all of whose Geodesics are Closed

0. Introduction.- A. Motivation and History.- B. Organization and Contents.- C. What is New in this Book?.- D. What are the Main Problems Today?.- 1. Basic Facts about the Geodesic Flow.- A.

Some new four-manifolds

The quotient space Q = X'/Z, of the involution is a smooth manifold of the (simple) homotopy type of real projective 4-space P', but not diffeomorphic or even piecewise linear (PL) homeomorphic to