Next: Discussion Up: Billiards that share a Previous: Devil's Staircase

Aubry-Mather Sets and the Invariant Circle

In this section, we analyze the interaction of the moving invariant circle tex2html_wrap_inline410 with Birkhoff periodic orbits of type p/q if the phase-locking interval tex2html_wrap_inline632 has a nonempty interior.

It is a basic fact for monotone twist maps that for any tex2html_wrap_inline768 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 tex2html_wrap_inline776 be the set of Birkhoff periodic orbits. Let tex2html_wrap_inline778 be irrational. An accumulation point (in the Hausdorff topology) of sets tex2html_wrap_inline776 as tex2html_wrap_inline782 is called an Aubry-Mather set and denoted by tex2html_wrap_inline784 . Such a set tex2html_wrap_inline784 has the property that it is the graph of a Lipschitz continuous function tex2html_wrap_inline788 and that a lift tex2html_wrap_inline790 of tex2html_wrap_inline462 preserves the order of the covering of M [5].

(In the following discussion, we fix the rotation number tex2html_wrap_inline796 . 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 tex2html_wrap_inline440 avoiding all fixed points of tex2html_wrap_inline806 . The index tex2html_wrap_inline808 of C with respect to tex2html_wrap_inline806 is defined as the Brower degree of the map tex2html_wrap_inline814 tex2html_wrap_inline816 . where tex2html_wrap_inline818 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 tex2html_wrap_inline806 .

If a curve C contains only one fixed point tex2html_wrap_inline826 of tex2html_wrap_inline806 , tex2html_wrap_inline830 is called the index of tex2html_wrap_inline826 . If C contains a connected fixed-point set tex2html_wrap_inline836 , we call tex2html_wrap_inline838 the index of this fixed point set. If a curve C contains finitely many fixed points tex2html_wrap_inline842 (rsp. connected fixed-point sets tex2html_wrap_inline844 ) of tex2html_wrap_inline806 , then tex2html_wrap_inline848 .

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 tex2html_wrap_inline852 parameterized by some interval I. Assume C is a simple closed curve in tex2html_wrap_inline440 such that for tex2html_wrap_inline860 , no fixed point of tex2html_wrap_inline862 is on C and for all tex2html_wrap_inline860 only finitely many connected fixed point sets are inside C. A parameter value tex2html_wrap_inline860 for which the number of connected components of fixed points of tex2html_wrap_inline862 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 tex2html_wrap_inline406 varies, the the invariant circle tex2html_wrap_inline410 moves through the phase space tex2html_wrap_inline440 . Sets in the region between the invariant circle and the boundary are called below tex2html_wrap_inline882 , the others are called above tex2html_wrap_inline882 .

For tex2html_wrap_inline406 near tex2html_wrap_inline888 , the invariant circle is near the boundary tex2html_wrap_inline890 of the phase space tex2html_wrap_inline440 . For tex2html_wrap_inline894 , the invariant circle tex2html_wrap_inline410 moves towards the equator tex2html_wrap_inline898 of tex2html_wrap_inline440 . Aubry-Mather sets tex2html_wrap_inline784 with a fixed rotation number tex2html_wrap_inline778 pass through the moving circle tex2html_wrap_inline410 .

The passage of tex2html_wrap_inline784 with irrational tex2html_wrap_inline778 is easy to describe: since each irrational tex2html_wrap_inline778 is a point of increase of tex2html_wrap_inline504 , the set of parameters tex2html_wrap_inline406 for which tex2html_wrap_inline784 intersects with tex2html_wrap_inline410 consists of exactly one parameter value tex2html_wrap_inline922 , for which tex2html_wrap_inline924 . The set tex2html_wrap_inline784 is in general a Cantor set, for tex2html_wrap_inline928 .

More interesting is when Birkhoff periodic orbits tex2html_wrap_inline930 pass through tex2html_wrap_inline410 if the phase locking interval tex2html_wrap_inline632 is nontrivial. Since any periodic orbit on tex2html_wrap_inline410 is a Birkhoff periodic orbit, tex2html_wrap_inline632 is the set of parameters tex2html_wrap_inline406 for which tex2html_wrap_inline930 intersects with tex2html_wrap_inline410 .

The following theorem is proven in [3] for a general convex caustic tex2html_wrap_inline402 .


The idea of the proof is as follows: for tex2html_wrap_inline978 and tex2html_wrap_inline980 , every Birkhoff periodic set on tex2html_wrap_inline410 is parabolic and has index -1. By the symmetry of the situation, it suffices to study the bifurcation at tex2html_wrap_inline986 . For tex2html_wrap_inline988 , there exists no Birkhoff periodic set on tex2html_wrap_inline410 and all such sets are above tex2html_wrap_inline410 . Local index conservation implies that such a set must exist nearby. Using properties of the one-parameter family of circle map tex2html_wrap_inline408 for tex2html_wrap_inline996 with tex2html_wrap_inline406 near tex2html_wrap_inline986 , we conclude that there exist 2 hyperbolic Birkhoff periodic sets of index -1 on tex2html_wrap_inline410 . 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 tex2html_wrap_inline1010 is on the other side of tex2html_wrap_inline410 .

tex2html_wrap1016 Fig. 11. Schematic illustration of the passage of the Birkhoff periodic orbits through the moving invariant curve tex2html_wrap_inline410 .
Next: Discussion Up: Billiards that share a Previous: Devil's Staircase
Oliver Knill, Jul 8, 1998