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Average linking numbers. (English) Zbl 0960.37006
This paper deals with the Hopf invariant for any invariant measure under a differentiable flow on \(S^3\). For any \(x\in S^3\) and \(T\in \mathbb{R}\) let \(\widehat{x}_T\) denote the knot formed by the orbit from \(x\) to \(\varphi_T(x)\) and a path \(\alpha_{\varphi_T(x),x}\) in a chosen system of short curves, joining \(\varphi_T(x)\) to \(x\), where \(\varphi_t\) is a given flow on \(S^3\). For any two points on different orbits \(p,q\) and times \(T_1, T_2\) define, when possible, their average linking number as \[ l(x,y):= \lim_{T_1,T_2\to \infty} \frac{1}{T_1T_2} l(\widehat{x}_T, \widehat{y}_T). \tag{1} \] The authors show that, under some natural assumptions this limit exists and does not depend on the set of short curves. Here \(l(\gamma_1,\gamma_2)\) is the standard linking number. In particular, they prove that for any invariant measure under differentiable flow \(\varphi_t\) on \(S^3\) without singularities, which has not periodic orbit of positive measure (1) always exists.

37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems
57M25 Knots and links in the \(3\)-sphere (MSC2010)
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
34C29 Averaging method for ordinary differential equations
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