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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.

MSC:
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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