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Asymptotic higher ergodic invariants of magnetic lines

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Topological Dynamics in the Physical and Biological Sciences

V.I.Arnol’d in 1984 formulated the following problem: “To transform asymptotic ergodic definition of Hopf invariant of a divergence-free vector field to Novikov’s theory, which generalizes Withehead product in homotopy groups”’.

We shall call divergence-free fields by magnetic fields. Asymptotic invariants of magnetic fields, in particular, the theorem by V.I.Arnol’d about asymptotic Gaussian linking number, is a bridge, which relates differential equitations and topology. We consider 3D case, the most important for applications.

Asymptotic invariants are derived from a finite-type invariant of links, which has to be satisfied corresponding limit relations. Ergodicity of such an invariant means that this invariant is well-defined as the mean value of an integrable function, which is defined on the finite-type configuration space $K$, associated with magnetic lines.

At the previous step of the construction we introduce a simplest infinite family of invariants: asymptotic linking coefficients. The definition of the invariants is simple: the helicity density is a well-defined function on the space $K$, the coefficients are well-defined as the corresponding integral momentum of this function. Using this general construction, a higher asymptotic ergodic invariant is well-defined. Assuming the the magnetic field is represented by a $delta$-support with contains 3 closed magnetic lines equipped with unite magnetic flows, this higher invariant is equal to the corresponding Vassiliev’s invariant of classical links of the order 7, and this invariant is not a function of the pairwise linking numbers of components. When the length of generic magnetic lines tends to $infty$, the asymptotic of the invariant is equal to 12, this is less then twice order $14$ of the invariant.

Preliminary results arXiv:1105.5876 was presented at the Conference ”`Entanglement and Linking”’ (Pisa) 18-19 May (2011).

This talk is part of the Isaac Newton Institute Seminar Series series.

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