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The question I`m not getting is the following:

$\alpha$ is irrational and $f: T^{2} \to T^{2}$ is the homeomorphic function from to 2-torus unto itself given by $f(x,y)=(x+\alpha,x+y)$.

$a )$ Prove that every non-empty, open, $f$-invariant set is dense.

My attempt:

I know I have to show that every orbit of this function is dense, but I don`t know how. Could you please help me?

Yours,

Pim

Pim
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    Just a comment: every orbit being dense is a stronger property than being topologically transitive - for instance, the full binary shift is topologically transitive but has many points whose orbits are not dense. You only need to show that there exists a single point with a dense orbit (for a torus anyway this is equivalent to be being topologically transitive). As with any question like this, I would suggest drawing a picture: what happens to the line whose point are labeled $(0,z)$? How about $(z,0)$, $(1,z)$, and $(z,1)$? – Dan Rust Oct 14 '16 at 15:33

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