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 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 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].

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.

Sign in / Sign up

Export Citation Format

Share Document