Let $A_\alpha$ be the affine map of the two-dimensional torus, i.e $$A_\alpha : T^2 \to T^2, \;\; A_\alpha (x,y) = (x + \alpha, y + x) \pmod 1.$$ Can someone give me a hint on how to find all invariant Borel probability measures for $A_\alpha$ when $\alpha$ is rational?
Thank you!
On
Here's a partial answer. Suppose, for example, $\alpha = \dfrac 3 7.$ Then you have $$ x\mapsto x + \frac 3 7 \mapsto x + \frac 6 7 \mapsto x + \frac 2 7 \mapsto x + \frac 5 7 \mapsto x + \frac 1 7 \mapsto x + \frac 4 7 \mapsto x \mapsto\cdots $$ This hits only finitely many points and then starts over. That means there are lots of open sets that it never reaches. And with any rational number, the number of steps it takes to return to where it started is the denominator of the rational number. Therefore ergodicity requires that $\alpha$ be irrational.
Since it is possible to write down explicitly a formula for the power $A_\alpha^n$, the best approach is to use Fourier series and show that there is uniform convergence for each character (as you probably know this is equivalent to unique ergodicity).
On your second question, again having an explicit formula is helpful, starting with the first component.