Fix an irrational number $\alpha$ and a smooth, positive, real function $p$ on the circle. A particle at a point $x$ in the circle jumps to $x+\alpha$ with probability $p(x)$ or to $x−\alpha$ with probability $1−p(x)$. In 1999 Sinai investigated ergodic properties of this Markov process for Diophantine $\alpha$. Throughout the talk I would like to present what I know about mixing properties when $\alpha$ is Liouville.
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