0
$\begingroup$

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.

Improve this question
$\endgroup$
1
  • $\begingroup$ 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. $\endgroup$
    – Andrey Gogolev
    Commented Sep 27, 2010 at 22:23

1 Answer 1

Reset to default
2
$\begingroup$

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 $\partial_u+t\partial_v.$

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .