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In this section, we analyze the interaction of the moving invariant
circle with Birkhoff periodic orbits of type *p*/*q* if
the phase-locking interval has a nonempty interior.

It is a basic fact for monotone twist maps that for any
at least two Birkhoff periodic orbits of type (*q*,*p*) exist
(see e.g. [7, 5]).
In the billiard case, one of these orbits is a local maximum of the
functional |*O*|, the total length of the trajectory *O*.
Let be the set of Birkhoff periodic orbits.
Let be irrational. An accumulation point
(in the Hausdorff topology) of sets
as is called an
*Aubry-Mather set* and denoted by . Such a set
has the property that it is the graph of a Lipschitz continuous function
and that a lift of
preserves the order of the covering of *M* [5].

(In the following discussion, we fix the rotation number .
Typically, Birkhoff periodic orbits are isolated. The case
of billiards shows however
that one has to deal in general with whole arcs of Birkhoff periodic orbits
If a connected set *Y* of periodic orbits
contains a Birkhoff periodic orbit, then every orbit in *Y* is a Birkhoff
periodic orbit and we will call such a set a *Birkhoff periodic set*.)

Let *C* be a simple, contractible closed curve in avoiding all
fixed points of . The *index*
of *C* with respect to is defined as
the Brower degree of the map
.
where is calculated in a chart.
The index is a homotopy invariant
and does not change if we deform *C* without intersecting
a fixed point of .

If a curve *C* contains only one fixed point of ,
is called the *index* of . If *C* contains
a connected fixed-point set , we call
the
*index of this fixed point set*.
If a curve *C* contains finitely
many fixed points (rsp. connected fixed-point sets )
of , then .

We will use the fact that the
index of a fixed point of an area-preserving
homeomorphism is bounded above
by 1 [10] [9].

Given a one-parameter
family of monotone twist maps
parameterized by some interval *I*.
Assume *C* is a simple closed curve in such that for ,
no fixed point of is on *C* and for all only finitely
many connected fixed point sets are inside *C*.
A parameter value for which
the number of connected components of fixed points of inside *C*
changes is called a *bifurcation parameter*.

Index considerations limit the possibilities for bifurcations
of periodic orbits in monotone twist maps.

As the parameter varies, the the invariant
circle
moves through the phase space .
Sets in the region between
the invariant circle and the boundary are called *below *,
the others are called *above *.

For near , the invariant circle is near the boundary
of the phase space .
For , the invariant circle moves
towards the
equator of .
Aubry-Mather sets with a fixed
rotation number pass through the moving circle .

The passage of
with irrational is easy to describe: since each irrational
is a point of increase of , the set of parameters
for which
intersects with consists of
exactly one parameter value ,
for which .
The set is in general a Cantor set, for
.

More interesting is when
Birkhoff periodic orbits pass through if
the phase locking interval is nontrivial.
Since any periodic orbit on is a Birkhoff periodic orbit,
is the set of parameters for which intersects
with .

The following theorem is proven in [3] for a general convex caustic .

The idea of the proof is as follows: for and , every Birkhoff periodic set on is parabolic and has index -1. By the symmetry of the situation, it suffices to study the bifurcation at . For , there exists no Birkhoff periodic set on and all such sets are above . Local index conservation implies that such a set must exist nearby. Using properties of the one-parameter family of circle map for with near , we conclude that there exist 2 hyperbolic Birkhoff periodic sets of index -1 on . Local index conservation implies that a Birkhoff periodic set of index +1 must be nearby. By the global Poincaré index formula, we argue that this set is on the other side of .

Fig. 11. Schematic illustration of the passage of the Birkhoff periodic orbits through the moving invariant curve . |