×

The trunkenness of a volume-preserving vector field. (English) Zbl 1381.37028

Topological invariants for vector fields \(X\) in compact three-dimensional manifolds – in particular, in \(S^3\) or in invariant compact regions \(B \subset \mathbb{R}^3\) – have long been actively sought for. This has a mathematical interest per se, but is also relevant in connection with fluid dynamics and magnetohydrodynamics.
If the vector field admits a closed orbit \(\gamma\), then knot invariants for \(\gamma\) provide a partial (in that only a small part of the field is taken into account) answer. More generally, one can consider the {helicity} \(\mathrm{Hel} (X)\).
Several attempts to find further regular invariants failed, in the sense that they led to invariants which turned out to be functions of helicity. It turns out this is unavoidably the case, as shown by A. Enciso et al. [Proc. Natl. Acad. Sci. USA 113, No. 8, 2035–2040 (2016; Zbl 1359.58006)], see also [E. A. Kudryavtseva, Math. Notes 99, No. 4, 611–615 (2016; Zbl 1422.58001); translation from Mat. Zametki 99, No. 4, 626–630 (2016) ; ibid. 95, No. 6, 877–880 (2014; Zbl 1370.37034); translation from Mat. Zametki 95, No. 6, 951–954 (2014)]. Roughly speaking, {any regular invariant} is indeed a function of helicity.
This theorem leaves open the possibility to find new invariants, provided they are not regular. The possibility considered in this paper is related to a knot invariant known as {trunkenness}, introduced by M. Ozawa [Geom. Dedicata 149, 85–94 (2010; Zbl 1220.57002)]; this requires that \(X\) preserves a measure \(\mu\), and is defined as \[ \mathrm{Tks} (X,\mu) \;= \;\inf_h \max_{t \in [0,1] } \, \mathrm{Flux} [X,\mu,h^{-1} (t)] \;= \;\inf_h \max_{t \in [0,1]} \, \lim_{\varepsilon \to 0} \, \frac{1}{\varepsilon} \, \mu [ \Phi_X^{[0,\varepsilon ]} (h^{-1} (t)) ] \;, \] where \(h\) are height functions and \(\Phi_X\) is the flow under \(X\).
The main results shown in this carefully written paper are as follows:
Theorem A. If \(X_1\) and \(X_2\) preserve \(\mu\) and there is a \(\mu\)-preserving homeomorphism \(f\) which conjugates their flows, then \(\mathrm{Tks} (X_1 , \mu)= \mathrm{Tks} (X_2, \mu)\).
Theorem B. The trunkenness is a continuous functional on the space of normal currents.
Corollary 1. If \((X_n ,\mu_n )\) is a sequence of measure-preserving vector fields with \(X_n \to X\) in the \(C^0\) topology and \(\mu_n \to \mu\) in the weak \(*\)-topology, then \(\lim_{n \to \infty} \mathrm{Tks} (X_n , \mu_n ) = \mathrm{Tks} (X , \mu )\).
Corollary 2. If \(X\) is \(\mu\)-preserving and ergodic w.r.t. \(\mu\), then for \(\mu\)-almost every loop \(k_X (p,t)\) starting from \(p\) and tangent to \(X\) for a time \(t\) (and then closed by a segment), the limit \(\lim_{t \to \infty} (1/t) \mathrm{Tks} (k_X (p,t))\) exists and is equal to \(\mathrm{Tks} (X,\mu)\).
Theorem C. There is no function \(f\) such that for every ergodic volume-preserving \(X\) on \(S^3\) it is \(\mathrm{Tks} (X,\mu) = f[ \mathrm{Hel} (X,\mu) ]\).
Theorem D. If \(X\) is a non-singular \(\mu\)-preserving vector field on \(S^3\) and \(h\) a height function such that \(\mathrm{Tks} (X,\mu) = \max_{t \in [0,1]} \mathrm{Flux} [X,\mu,h^{-1} (t)]\), then \(X\) has an unknotted periodic orbit.

MSC:

37C10 Dynamics induced by flows and semiflows
37C05 Dynamical systems involving smooth mappings and diffeomorphisms
37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems
37C27 Periodic orbits of vector fields and flows
34C14 Symmetries, invariants of ordinary differential equations
34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
76W05 Magnetohydrodynamics and electrohydrodynamics
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Arnold Vladimir I 1974 The asymptotic Hopf invariant and its applications Proc. Summer School in Differential Equations at Dilizhan, 1973 Evevan (in Russian)
[2] Arnold Vladimir I 1986 The asymptotic Hopf invariant and its applications Sel. Math. Sov.5 327-45 English transl. · Zbl 0623.57016
[3] Arnold Vladimir I and Khesin B 1998 Topological Methods in Hydrodynamics (Berlin: Springer)
[4] Baader S 2011 Asymptotic concordance invariants for ergodic vector fields Comment. Math. Helv.86 1-12 · Zbl 1209.57007
[5] Baader S and Marché J 2012 Asymptotic Vassiliev invariants for vector fields Bull. Soc. Math. France140 569-82 · Zbl 1278.57017
[6] Enciso A, Peralta-Salas D and Torres de Lizaur F 2016 Helicity is the only integral invariant of volume-preserving transformations Proc. Natl Acad. Sci. USA113 2035-40 · Zbl 1359.58006
[7] Freedman M H and He Z-X 1991 Divergence-free fields: energy and asymptotic crossing number Ann. Math.134 189-229 · Zbl 0746.57011
[8] Gabai D 1987 Foliations and the topology of 3-manifolds III J. Differ. Geom.26 479-536 · Zbl 0639.57008
[9] Gambaudo J-M and Ghys É 2001 Signature asymptotique d’un champ de vecteurs en dimension 3 Duke Math. J.106 41-79 · Zbl 1010.37010
[10] von Helmholtz H 1858 Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen J. Reine Angew. Math.55 25-55 · ERAM 055.1448cj
[11] Katok A B 1973 Ergodic perturbations of degenerate integrable Hamiltonian systems Math. USSR Izv.7 535 · Zbl 0316.58010
[12] Kudryavtseva E A 2014 Conjugation invariants on the group of area-preserving diffeomorphisms of the disk Math. Notes95 877-80 · Zbl 1370.37034
[13] Kudryavtseva E A 2016 Helicity is the only invariant of incompressible flows whose derivative is continuous in C1-topology Math. Notes99 611-5 · Zbl 1422.58001
[14] Kuperberg K 1994 A smooth counterexample to the Seifert conjecture Ann. Math.140 723-32 · Zbl 0856.57024
[15] Kuperberg G 1996 A volume-preserving counterexample to the Seifert conjecture Comment. Math. Helv.71 70-97 · Zbl 0859.57017
[16] Moffatt K 1969 The degree of knottedness of tangle vortex lines J. Fluid. Mech.106 117-29 · Zbl 0159.57903
[17] Moreau J-J 1961 Constantes d’un îlot tourbillonnaire en fluide parfait barotrope C. R. Acad. Sci. Paris252 2810-2 · Zbl 0151.41703
[18] Morgan F 2000 Geometric Measure Theory. A Beginner’s Guide 3rd edn (San Diego, CA: Academic)
[19] Ozawa M 2010 Waist and trunk of knots Geom. Dedicata149 85-94 · Zbl 1220.57002
[20] Tait P G 1877 On knots Trans. R. Soc. Edinburgh28 145-90 · JFM 09.0392.09
[21] Thomson W 1867 On vortex atoms Proc. R. Soc. Edinburgh6 94-105
[22] Thomson W 1867 On vortex atoms Phil. Mag.34 15-24 reprinted in
[23] Vogel T 2002 On the asymptotic linking number Proc. Am. Math. Soc.131 2289-97 · Zbl 1015.57018
[24] Woltjer L 1958 A theorem on force-free magnetic fields Proc. Natl Acad. Sci. USA44 489-91 · Zbl 0081.21703
[25] Zupan A 2012 A lower bound on the width of satellite knots Topol. Proc.40 179-88 · Zbl 1280.57012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.