Aubry-Mather Sets and the Invariant Circle

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 .