Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
    
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
add comment

1 Answer

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 $\partial_u+t\partial_v.$

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.