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A symbolic dynamics system. Pick 2 (real number) parameters a (phase) and w (frequency). Now define xi to be 1 if sin(a + i w) ≥ 0 and 0 otherwise.

Think of a wheel spinning at angular velocity w, with a spot marked at angle a. xi says if the spot is on the left or right half at time i.

Now take the set of all shifts Snx, and take the closure of that (since all systems in symbolic dynamics are closed sets). Call it Ka,w; this is the Kronecker system. Kronecker's lemma is closely related; it says that if w is irrational, then the set of all values of sin(i w) is dense.

If w is rational, then the system is finite and x is periodic; that's not the interesting case. When w is irrational, it turns out that (due to Kronecker's lemma) the set Ka,w doesn't depend on a. In fact, if we take a=0, just one point has to be added to the orbit of x to get a closed set: we chose (arbitrarily) to take xi=1 when sin(i w)=0, but we could equally well have chosen to take 0 in that case. This "alternative" point is the only point on the boundary of the orbit of x.

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