MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $f:M\rightarrow M$ be a $C^2$ hyperbolic diffeomorphism on compact connected riemannian manifold $M$. then there are local stable and unstable manifolds at each point denoted by $W^s_\delta(x), W^u_\delta(x)$ for sufficiently small $\delta>0$. its well known that $W^s(x)=\cup_{n\geq0}f^{-n}(W^s_\delta(f^n(x))) , W^u(x)=\cup_{n\geq0}f^{n}(W^u_\delta(f^{-n}(x)))$ are absolutely continuous foliations on $M$.

An argument which i have met several times is that $\cup_{x\in W^u_\delta(y)}W^s_\delta(x)$ contains an open ball $B(y, r)$ for some $r>0$.I can see this in a geometric interpretation but i can't proof it. i was wondering if someone could help me please.

share|cite|improve this question
up vote 4 down vote accepted

The local product structure says that for hyperbolic diffeos (it works also for flows), given two points $z$ and $z'$ in a small neighbourhood of $y$, then $W^s_\delta(z)$ and $W^u_\delta(z')$ intersect at exactly one point, often denoted by $[z,z']$.

Apply this property on $B(y,r)$ for $r$ small enough to any $z\in B(y,r)$ and $z'=y$. Then $W^u_\delta(y)$ and $W^s_\delta(z)$ intersect at exactly one point, that you call $x$. It proves the desired property.


share|cite|improve this answer

Your Answer


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.