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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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