## |

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 . |

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