We propose a definition of a higher asymptotic ergodic invariant of finite type for a divergence-free vector field in a three-dimensional ball and partially construct an example of such an invariant. Bibliography: 6 titles.
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P. M. Akhmet’ev, “Quadratic helicities and the energy of magnetic fields” [in Russian], Tr. Mat. Inst. Steklova 278, 16–28 (2012); English transl.: Proc. Steklov Inst. Math. 278, 10–21 (2012).
P. M. Akhmet’ev, “On a higher integral invariant for closed magnetic lines,” J. Geom. Phys. 74, 381–391 (2013).
P. M. Akhmet’ev, “On a new integral formula for an invariant of 3-component oriented links,” J. Geom. Phys. 53, 180–196 (2005).
P. M. Akhmet’ev and O. V. Kunakovskaya, “Integral formula for a generalized Sato–Levine invariant in magnetic hydrodynamics,” Mat. Zametki 85, No. 4, 524–537 (2009); English transl.: Math. Notes 85, No. 4, 503–514 (2009).
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Translated from Problemy Matematicheskogo Analiza 79, March 2015, pp. 25–34.
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Akhmet’ev, P.M. On Arnold’s Problem on Higher Analogs of the Asymptotic HOPF Invariant. J Math Sci 208, 24–35 (2015). https://doi.org/10.1007/s10958-015-2420-7
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DOI: https://doi.org/10.1007/s10958-015-2420-7