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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?

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Not a priori. This would immediately imply transitivity which is an open problem

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