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

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

Now take the set of all shifts **S**^{n}x, and take the closure of that (since all systems in symbolic dynamics are closed sets). Call it **K**_{a,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 **K**_{a,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 *x*_{i}=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*.