Next: Aubry-Mather Sets and the Up: Billiards that share a Previous: A Family of Circle

# Devil's Staircase

The graph of a nondecreasing function, , is called a devil's staircase if there exists a family of disjoint open intervals, such that their union is dense in , the function is constant on each interval and such that takes different values on different intervals (see, e.g., [6, 8]).

A rational number, is called exceptional, if the homeomorphism such that is conjugate to . For any non-exceptional p/q, the set is an interval, with nonempty interior. Using the terminology of circle maps, we call a phase locking interval. If p/q is exceptional, the interval degenerates to a single point . In this case, every point on the invariant curve is periodic. From the formula in Lemma 2.2 of we get:

This leads to the following Corollary:

By differentiating this relation along a deformation, we get necessary conditions which have to hold in an exceptional case.

Let be a q-periodic orbit, and let be a deformation of O. We assume that I is an interval with a nonempty interior, that . Let be the polygon corresponding to O(t). We will assume that the deformation O(t) is differentiable for and nontrivial. By this we mean that are -functions, that for , and that at least one for . We say a periodic orbit admits a deformation if O is contained in an interval of periodic orbits.

Proof. Assume a deformation O(t) exists. Let be parameterized by arc lengths . We have and . Use this and the mirror equation of geometrical optics to get

Differentiating obtained from Equation (1) gives

This is not possible since for at least one j and for all . End of the proof.

 Fig. 9. Proof of Lemma 3.3.

This is a special case of a more general theorem, which we have proven in [3]. There is first of all a geometrical argument which shows that for a generic convex curve , the rotation function is a devil's staircase. We also showed that if has a flat point of if is a polygon, then the rotation function is a devil's staircase.

Figure 10 shows a numerically computed graph of the rotation function in our case, when the is an equilateral triangle. The graph shows the rotation number in dependence on .

 Fig. 10. Numerical computation of the rotation number in dependence of . The plateau at the beginning corresponds to rotation number 1/3. The end of the initial plateau is given by the value .

Next: Aubry-Mather Sets and the Up: Billiards that share a Previous: A Family of Circle

Oliver Knill, Wed Jul 8, 1998