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
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 .|
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