A pasting lemma and some applications for conservative systems

2007 ◽  
Vol 27 (5) ◽  
pp. 1399-1417 ◽  
Author(s):  
ALEXANDER ARBIETO ◽  
CARLOS MATHEUS
Keyword(s):  
Vector Field ◽  
Compact Manifold ◽  
Vector Fields ◽  
Three Dimensional ◽  
Invariant Set ◽  
Dominated Splitting ◽  
Divergence Free Vector ◽  
Conservative Systems ◽  
Divergence Free ◽  
Volume Preserving

AbstractWe prove that in a compact manifold of dimension n≥2, C1+α volume-preserving diffeomorphisms that are robustly transitive in the C1-topology have a dominated splitting. Also we prove that for three-dimensional compact manifolds, an isolated robustly transitive invariant set for a divergence-free vector field cannot have a singularity. In particular, we prove that robustly transitive divergence-free vector fields in three-dimensional manifolds are Anosov. For this, we prove a ‘pasting’ lemma, which allows us to make perturbations in conservative systems.

scholarly journals Vector fields satisfying the barycenter property

2018 ◽  
Vol 16 (1) ◽  
pp. 429-436 ◽  
Author(s):  
Manseob Lee
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Axiom A ◽  
Divergence Free Vector ◽  
Divergence Free ◽  
Volume Preserving

AbstractWe show that if a vector fieldXhas theC1robustly barycenter property then it does not have singularities and it is Axiom A without cycles. Moreover, if a genericC1-vector field has the barycenter property then it does not have singularities and it is Axiom A without cycles. Moreover, we apply the results to the divergence free vector fields. It is an extension of the results of the barycenter property for generic diffeomorphisms and volume preserving diffeomorphisms [1].

The Lyapunov exponents of generic zero divergence three-dimensional vector fields

2007 ◽  
Vol 27 (5) ◽  
pp. 1445-1472 ◽  
Author(s):  
MÁRIO BESSA
Keyword(s):  
Vector Field ◽  
Lyapunov Exponents ◽  
Vector Fields ◽  
Dense Subset ◽  
Three Dimensional ◽  
Invariant Set ◽  
Compact Manifolds ◽  
Dimensional Vector ◽  
Dominated Splitting

AbstractWe prove that for a C1-generic (dense Gδ) subset of all the conservative vector fields on three-dimensional compact manifolds without singularities, we have for Lebesgue almost every (a.e.) point p∈M that either the Lyapunov exponents at p are zero or X is an Anosov vector field. Then we prove that for a C1-dense subset of all the conservative vector fields on three-dimensional compact manifolds, we have for Lebesgue a.e. p∈M that either the Lyapunov exponents at p are zero or p belongs to a compact invariant set with dominated splitting for the linear Poincaré flow.

scholarly journals On volume-preserving vector fields and finite-type invariants of knots

2014 ◽  
Vol 36 (3) ◽  
pp. 832-859 ◽  
Author(s):  
R. KOMENDARCZYK ◽  
I. VOLIĆ
Keyword(s):  
Finite Type ◽  
Configuration Space ◽  
Vector Fields ◽  
Finite Type Invariants ◽  
Linking Number ◽  
Divergence Free Vector ◽  
Divergence Free ◽  
Volume Preserving ◽  
Configuration Space Integrals

We consider the general non-vanishing, divergence-free vector fields defined on a domain in$3$-space and tangent to its boundary. Based on the theory of finite-type invariants, we define a family of invariants for such fields, in the style of Arnold’s asymptotic linking number. Our approach is based on the configuration space integrals due to Bott and Taubes.

scholarly journals Helicity is the only integral invariant of volume-preserving transformations

2016 ◽  
Vol 113 (8) ◽  
pp. 2035-2040 ◽  
Author(s):  
Alberto Enciso ◽  
Daniel Peralta-Salas ◽  
Francisco Torres de Lizaur
Keyword(s):  
Vector Fields ◽  
Integral Kernel ◽  
Integral Invariant ◽  
Divergence Free Vector ◽  
Divergence Free ◽  
Volume Preserving ◽  
Arbitrary Volume

We prove that any regular integral invariant of volume-preserving transformations is equivalent to the helicity. Specifically, given a functional ℐ defined on exact divergence-free vector fields of class C1 on a compact 3-manifold that is associated with a well-behaved integral kernel, we prove that ℐ is invariant under arbitrary volume-preserving diffeomorphisms if and only if it is a function of the helicity.

scholarly journals Representation of vector fields

2015 ◽  
Vol 3 (2) ◽  
pp. 73
Author(s):  
Alexander G. Ramm
Keyword(s):  
Vector Field ◽  
Explicit Formula ◽  
Simple Proof ◽  
Vector Fields ◽  
Smooth Domain ◽  
Gradient Field ◽  
Smooth Vector ◽  
Smooth Vector Field ◽  
Divergence Free Vector ◽  
Divergence Free

<p>A simple proof is given for the explicit formula which allows one to recover a \(C^2\) – smooth vector field \(A=A(x)\) in \(\mathbb{R}^3\), decaying at infinity, from the knowledge of its \(\nabla \times A\) and \(\nabla \cdot A\). The representation of \(A\) as a sum of the gradient field and a divergence-free vector fields is derived from this formula. Similar results are obtained for a vector field in a bounded \(C^2\) - smooth domain.</p>

Unitary vector fields are Fermi–Walker transported along Rytov–Legendre curves

2015 ◽  
Vol 12 (10) ◽  
pp. 1550111 ◽  
Author(s):  
Mircea Crasmareanu ◽  
Camelia Frigioiu
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Killing Vector ◽  
Three Dimensional ◽  
Ricci Soliton ◽  
Conformal Killing ◽  
Space Forms ◽  
Potential Vector ◽  
Legendre Curves ◽  
Complex Space Forms

Fix ξ a unitary vector field on a Riemannian manifold M and γ a non-geodesic Frenet curve on M satisfying the Rytov law of polarization optics. We prove in these conditions that γ is a Legendre curve for ξ if and only if the γ-Fermi–Walker covariant derivative of ξ vanishes. The cases when γ is circle or helix as well as ξ is (conformal) Killing vector filed or potential vector field of a Ricci soliton are analyzed and an example involving a three-dimensional warped metric is provided. We discuss also K-(para)contact, particularly (para)Sasakian, manifolds and hypersurfaces in complex space forms.

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