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 .
(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  .
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
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
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
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
The following theorem is proven in  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 .|