 Irrational skew shift

For irrational $\alpha$, the irrational skew shift is the skew product $T_{\alpha }(x,y)=(x+\alpha ,y+x)$ on the torus $(\mathbb{R}/\mathbb{Z}{)}^2$. It preserves normalized Lebesgue measure and has iterates $T_{\alpha }^n(x,y)=(x+n\alpha ,y+nx+\alpha n(n-1)/2)$ modulo one. Its Koopman operator sends the Fourier basis character $e_{r,s}=e^{2\pi i(rx+sy)}$ to $e^{2\pi ir\alpha }{e}_{r+s,s}$. The resulting infinite chains of Fourier coefficients show that it is an ergodic transformation. Moreover, uniform equidistribution of an irrational skew shift proves unique ergodicity.