= Irrational skew shift
{title2=$T_\alpha(x,y)=(x+\alpha,y+x)\pmod1$}
= Irrational skew translation
{synonym}
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>.
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