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Devil's Staircase

The graph of a nondecreasing function, tex2html_wrap_inline610 , is called a devil's staircase if there exists a family of disjoint open intervals, such that their union is dense in tex2html_wrap_inline612 , the function tex2html_wrap_inline414 is constant on each interval and such that tex2html_wrap_inline414 takes different values on different intervals (see, e.g., [6, 8]).

A rational number, tex2html_wrap_inline618 is called exceptional, if the homeomorphism tex2html_wrap_inline408 such that tex2html_wrap_inline604 is conjugate to tex2html_wrap_inline624 . For any non-exceptional p/q, the set tex2html_wrap_inline628 is an interval, tex2html_wrap_inline630 with nonempty interior. Using the terminology of circle maps, we call tex2html_wrap_inline632 a phase locking interval. If p/q is exceptional, the interval tex2html_wrap_inline632 degenerates to a single point tex2html_wrap_inline638 . In this case, every point on the invariant curve tex2html_wrap_inline410 is periodic. From the formula in Lemma 2.2 of tex2html_wrap_inline408 we get:

   lemma157

This leads to the following Corollary:

   coro167

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

Let tex2html_wrap_inline676 be a q-periodic orbit, and let tex2html_wrap_inline680 be a deformation of O. We assume that I is an interval with a nonempty interior, that tex2html_wrap_inline686 . Let tex2html_wrap_inline688 be the polygon corresponding to O(t). We will assume that the deformation O(t) is differentiable for tex2html_wrap_inline694 and nontrivial. By this we mean that tex2html_wrap_inline696 are tex2html_wrap_inline698 -functions, that tex2html_wrap_inline700 for tex2html_wrap_inline702 , and that at least one tex2html_wrap_inline704 for tex2html_wrap_inline694 . We say a periodic orbit tex2html_wrap_inline676 admits a deformation if O is contained in an interval of periodic orbits.

  lemma186

Proof. Assume a deformation O(t) exists. Let tex2html_wrap_inline716 be parameterized by arc lengths tex2html_wrap_inline718 . We have tex2html_wrap_inline720 and tex2html_wrap_inline722 . Use this and the mirror equation of geometrical optics to get

displaymath724

Differentiating tex2html_wrap_inline726 obtained from Equation (1) gives

displaymath728

This is not possible since tex2html_wrap_inline704 for at least one j and tex2html_wrap_inline734 for all tex2html_wrap_inline702 . End of the proof.

tex2html_wrap758 Fig. 9. Proof of Lemma 3.3.

coro205

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 tex2html_wrap_inline402 , the rotation function is a devil's staircase. We also showed that if tex2html_wrap_inline402 has a flat point of if tex2html_wrap_inline402 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 tex2html_wrap_inline402 is an equilateral triangle. The graph shows the rotation number in dependence on tex2html_wrap_inline406 .

tex2html_wrap760 Fig. 10. Numerical computation of the rotation number in dependence of tex2html_wrap_inline406 . The plateau at the beginning corresponds to rotation number 1/3. The end of the initial plateau is given by the value tex2html_wrap_inline756 .

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

Oliver Knill, Wed Jul 8, 1998