copies of the unit circle and mark off segment of length in the counterclockwise direction on each one with endpoint at 0. Now take irrational and consider the following dynamical system. Start with a point , say in the first circle. Rotate counterclockwise by until the first time the orbit lands in ; then switch to the corresponding point in the second circle, rotate by until the first time the point lands in ; switch back to the first circle and so forth. Veech showed that if is irrational, then there exists irrational for which this system is minimal and the Lebesgue measure is not uniquely ergodic." See also Bernoulli map Modular arithmetic Siegel disc Toeplitz algebra Phase locking (circle map) References Further reading C. E. Silva, Invitation to ergodic theory, Student Mathematical Library, vol 42, American Mathematical Society, 2008 Dynamical systems Irrational numbers