Let $(M, g)$ be a compact, connected Riemann without boundary and let $f: M\rightarrow M$ be Anosov. The nonwandering set of $f$, $\Omega (f)$, has a decomposition into finitely many closed, invariant, "basic" sets 


$\Omega (f) =B\_1 \;\cup \;... \;\cup \;B\_n$


So that $f|\_{B\_i}$ is topologically transitive. We also have 


$M = \bigcup\_{i=1}^n W^s(B\_i) = \bigcup\_{i=1}^n W^u(B\_i)$, 


where $W^s(B\_i):= \bigcup\_{x\in B\_i}W^s(x)$, $W^u(B\_i):= \bigcup\_{x\in B\_i}W^u(x)$, where $W^s(x)$ is the global stable manifold of $x$ and similarly for the unstable manifold.


Given two distinct basic sets $B\_i, B\_j$ is one of


$W^s(B\_i) \cap W^u(B\_j)$


$W^u(B\_i) \cap W^s(B\_j)$


always nonempty? 



Is there some point of intersection between the global stable and unstable manifolds of basic sets of an Anosov diffeomorphism?
Not a priori. This would immediately imply transitivity which is an open problem 


