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I would like to know an example of a compact Riemannian manifold $M$ and a smooth vector field $X$ on $M$ such that the flow $F$ associated to $X$ is defined for all time and for some vector $v\in T_pM$, the norm $$|dF_t(v)|\rightarrow \infty$$ as $t\rightarrow\infty$. Thanks.

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Any flow that admits an expanding sub-bundle will do. Anosov flows for example. Concrete examples: geodesic flow on a manifold of negative curvature, suspension flow of a toral automorphism that has at least one eigenvalue of modulus >1. – Andrey Gogolev Sep 27 '10 at 22:23
up vote 2 down vote accepted

For a minimal example, take $M:=\mathbb{S}^1\times \mathbb{S}^1$ where $\mathbb{S}^1:=\mathbb{R}/2\pi \mathbb{Z},$ with the field $X(u,v):=\sin(u)\partial_v.$ Then $F\\ ^t(u,v)=\big(u,v+\sin(u)t\big)$ and the differential of the flow at time $t$, computed on the vector $\partial_u$ at $p=(0,0)$ is just $\partial_u+t\partial_v.$

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