I have the following question : given some Anosov $T : \mathbb{T}^2 \to \mathbb{T}^2$ preserving the Lebesgue measure on the torus, is it true that for any arbitrary $x, y \in \mathbb{T}^2$, we have that $W^s(x) \cap W^u(y) \neq \emptyset$ ? I know that locally, this is true, so I suspect that (since $\mathbb{T}^2$ is connected) it remains true globally...
Help grealty appreciated !
Much more is true, namely $W^s(x)$ and $W^u(y)$ are each dense on $\mathbb T^2$, and their intersection $W^s(x) \cap W^u(y)$ is also dense.
To prove this, one can use the theorem (I think due to Roy Adler) which says that $T$ is topologically conjugate to a linear Anosov automorphism, for which the properties I wrote are more or less obvious: the stable foliation has a constant irrational slope, and the unstable foliation has a different constant irrational slope.
