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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.
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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 
.
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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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