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# Aubry-Mather sets and the invariant circle

In this section, we analyze the interaction of the moving invariant circle with the Birkhoff periodic orbits of type p/q, if the phase-locking interval is nontrivial.

Birkhoff periodic orbits and Aubry-Mather sets.
Denote by a lift of to and let R be the map . A periodic orbit is called a (p,q)-Birkhoff periodic orbit, if there exists a continuous function such that . It follows from a basic fact for monotone twist maps that for any at least two Birkhoff periodic orbits of type (p,q) exist [18, 15, 11]. One of these orbits is a maximum of the functional |O|, the total length of the trajectory O.

Let be the set of Birkhoff periodic orbits. Given 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 defined on some compact subset K of and such that a lift of preserves the order of the covering of M [15].

Typically, Birkhoff periodic points are isolated. The case of billiards shows however that one has to deal in general with whole arcs of Birkhoff periodic points. If a connected periodic set Y contains a Birkhoff periodic point, then every point in Y is a Birkhoff periodic point and we will call a connected component a Birkhoff periodic set. In a typical situation, a Birkhoff periodic set is a Birkhoff periodic point. The set of Birkhoff periodic points is isolated from the rest of (p,q)-periodic orbits.

The index of a periodic set.
Let C be a simple, contractible oriented closed curve in avoiding all fixed points of . The index of C with respect to the fixed points of is defined as the Brower degree of the map , , where is calculated in a chart. The index does not change if we deform C without intersecting a fixed point of . The index is also a homotopy invariant: if is a family of maps and for all in some parameter interval I and C is disjoint from the fixed points of , then is constant for .

If a curve C contains only one fixed point of , then 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 (see for example [5] Theorem 14.4.4 or [25]).

Examples. Assume C contains only one hyperbolic periodic point of period q, then . If C contains one elliptic periodic point, then . If C contains no periodic point, then . More generally: if is differentiable and for a fixed point of the matrix has no eigenvalue 1, then has index 1 if and index -1 if . (See [23] Lemma 4 or [28] Proposition 3.). The index can geometrically also be determined by Poincaré's index formula (see [28]). The index of a fixed point of an area-preserving homeomorphism is bounded above by 1 [28] [25]. From this follows by a generalized Poincaré-Hopf's theorem [27] that if the induced homeomorphism on has n attracting periodic intervals and so n repelling intervals on (they all must have index -1), then there exist n sets of index 1 of the same period on each side of .

Remark. A periodic orbit of a differentiable monotone twist map is a critical point of a functional. If the critical point is nondegenerate, then the orbit is either elliptic or hyperbolic. The Morse-index, the number of negative eigenvalues at a critical point and the index are related by . The reason is that a nondegenerate hyperbolic orbit has even Morse-index and a nondegenerate elliptic orbit has odd Morse-index (see [32]).

Bifurcations of sets of periodic orbits.
We consider a one-parameter family of monotone twist maps parameterized by points in some interval I. Let C be a homotopically trivial simple closed curve in such that for , no fixed point of is on C and such that C is the boundary of an open subset of . Since is bounded away from 0 and Equation 2), there are only finitely many connected fixed point sets in . A parameter value , for which the number of connected components of fixed points of of in changes, is called a bifurcation point. (Usually, a bifurcation point is a parameter point, for which the topological type of the fixed point set inside C changes. Since here, only the number of components will be relevant, we take the narrower definition.) Index considerations limit the possibilities for bifurcations of periodic orbits in monotone twist maps.

Examples. 1) Assume, for each , there exists at least one fixed point of in . Assume is hyperbolic for , parabolic for and elliptic for . If is the only fixed point in for , then is a bifurcation value because the index of changes from -1 to 1 and the sum of the indices of all fixed points of in stays constant. If two additional fixed points of index -1 are created at , then the bifurcation is a pitchfork bifurcation.
2) Assume, a single fixed point exists in for , and does not exist for . Then it must have index 0 and must be parabolic.
3) Also the reversed processes are possible: for example, two fixed points of index -1 collide with a fixed point of index 1 and leave a fixed point of index -1, or three fixed points, two of them with index -1 and one with index 1, collide for leaving a lonely fixed point of index -1.

Remarks. 1) If all periodic orbits of rotation number p/q are isolated, then at least two Birkhoff periodic orbits have index different from 0 because any periodic orbit of index 0 can be removed by a local perturbation of the map [30].
2) The bifurcations of periodic points of a generic one-parameter family of area-preserving maps of a surface has been classified in [22].

Passage of Aubry-Mather sets through the invariant circle.
As the parameter varies, the invariant circle belonging to the caustic moves up in the phase space . At each parameter , the invariant circle divides the phase space into a region below bounded by and and a region above .

For small , the invariant circle is near the boundary component of the phase space . For , the invariant circle approaches 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 parameter value with irrational is a point of increase of , the set of parameters for which intersects with consists of exactly one point, the parameter value , for which . The set is in general a Cantor set.

If the map and the curve are smooth and satisfies a Diophantine condition, then more can be said: both and depend in a differentiable way on (see [4]). If is smooth and is a smooth invariant curve with Diophantine rotation number , then is accumulated by other invariant curves filling a set of positive measure [33]. Therefore, if is Diophantine, the moving circle is accumulated by invariant curves which fill out a set of positive measure.

We analyze, how M(p/q) which consists of Birkhoff periodic orbits passes 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 M(p/q) intersects .

The picture is that for a residual set of caustics and a dense set of parameters , every Birkhoff periodic orbit on the invariant curve is isolated, hyperbolic and has index -1. Since these orbits correspond to periodic orbits of the circle map , we can give them coordinates and group them in pairs . (Compare Figure 11a). Important for us is the following Corollary.

Every q-periodic orbit of the billiard map is determined by the values and is a critical point of the length functional , which is the length of the polygon.

Note. We used here again the notation , but for different things. Since there is no danger of confusion and this notation is common for the entries of a Jacobi matrix.

The passage of Birkhoff periodic sets of a given rotation number p/q is in general a complicated sequence of possibly simultaneous bifurcations. There are two basic building blocks, the passage of an index -1 set and the passage of an index +1 set. Both events are bifurcations and the second event is the time reversed process of the first one. We first look at the first rsp. last bifurcation value.

The same argument proves also:

Remarks. 1) Let us consider the motion of the non-Birkhoff periodic points of rotation number p/q. By a theorem of Boyland and Hall [3], they exist for p/q sufficiently close to an irrational , if there is no invariant circle with rotation number . For , the non-Birkhoff periodic points can only be above and for , they can exist only below . On the other hand, they can not collide with . Thus, the non-Birkhoff periodic orbits appear and disappear in the course of the deformation.
2) We believe that typically, for sufficiently near to , an isolated hyperbolic Birkhoff periodic orbit of index -1 approaches the caustic, hits the caustic as a parabolic orbit of index -1 and leaves the caustic as an isolated elliptic orbit with index 1. At the point , two hyperbolic orbits of index -1 are created and move along the caustic. They might interact with other orbits of index -1 and each of those events is accompanied with a passage of a Birkhoff periodic orbit through the caustic as described in Theorem 5.6. At the end of the interval , the last Birkhoff periodic orbit of index 1 passes through the caustic, changing from elliptic to parabolic and leaves it hyperbolic with index -1.
3) If the phase locking interval degenerates to one point , then all the Birkhoff periodic sets hit the curve at the same time and is exceptional consisting of periodic points.
4) Since by Lemma 2.1, the circle diffeomorphisms have derivatives bounded away from zero and , there are only finitely many bifurcation points in . There are the following conservation laws for collisions of periodic sets with the invariant curve belonging to the caustic: to each orbit with index 1 hitting the curve for some , there is a set of index 1 leaving for some . The same is true for orbits with index -1. Thus, the sum of the indices of Birkhoff periodic orbits passing through the caustic curve is equal to zero.
5) The pitchfork bifurcations (see Figure 13) of the Birkhoff periodic orbit in the moments of intersection with the caustic correspond to saddle-node bifurcations of the circle map.

Next: On this document... Up: Billiards that share a Previous: Periodic orbits and

Oliver Knill
Wed Jul 8 11:57:32 CDT 1998