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Aubry-Mather sets and the invariant circle

In this section, we analyze the interaction of the moving invariant circle tex2html_wrap_inline1384 with the Birkhoff periodic orbits of type p/q, if the phase-locking interval tex2html_wrap_inline1586 is nontrivial.

Birkhoff periodic orbits and Aubry-Mather sets.
Denote by tex2html_wrap_inline2670 a lift of tex2html_wrap_inline1414 to tex2html_wrap_inline2674 and let R be the map tex2html_wrap_inline2678 . A periodic orbit is called a (p,q)-Birkhoff periodic orbit, if there exists a continuous function tex2html_wrap_inline2680 such that tex2html_wrap_inline2682 . It follows from a basic fact for monotone twist maps that for any tex2html_wrap_inline2684 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 tex2html_wrap_inline1400 be the set of Birkhoff periodic orbits. Given tex2html_wrap_inline2694 irrational. An accumulation point (in the Hausdorff topology) of sets tex2html_wrap_inline1400 as tex2html_wrap_inline2698 is called an Aubry-Mather set and denoted by tex2html_wrap_inline1394 . Such a set tex2html_wrap_inline1394 has the property that it is the graph of a Lipschitz continuous function tex2html_wrap_inline2704 defined on some compact subset K of tex2html_wrap_inline2708 and such that a lift tex2html_wrap_inline2670 of tex2html_wrap_inline1414 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 tex2html_wrap_inline1422 avoiding all fixed points of tex2html_wrap_inline2726 . The index tex2html_wrap_inline2728 of C with respect to the fixed points of tex2html_wrap_inline2726 is defined as the Brower degree of the map tex2html_wrap_inline2734 , tex2html_wrap_inline2736 , where tex2html_wrap_inline2738 is calculated in a chart. The index does not change if we deform C without intersecting a fixed point of tex2html_wrap_inline2726 . The index is also a homotopy invariant: if tex2html_wrap_inline1374 is a family of maps and for all tex2html_wrap_inline1378 in some parameter interval I and C is disjoint from the fixed points of tex2html_wrap_inline2726 , then tex2html_wrap_inline2754 is constant for tex2html_wrap_inline2756 .

If a curve C contains only one fixed point tex2html_wrap_inline2760 of tex2html_wrap_inline2726 , then tex2html_wrap_inline2764 is called the index of tex2html_wrap_inline2760 . If C contains a connected fixed-point set tex2html_wrap_inline2770 , we call tex2html_wrap_inline2772 the index of this fixed point set. If a curve C contains finitely many fixed points tex2html_wrap_inline2776 (rsp. connected fixed-point sets tex2html_wrap_inline2778 ) of tex2html_wrap_inline2726 , then tex2html_wrap_inline2782 (see for example [5] Theorem 14.4.4 or [25]).

Examples. Assume C contains only one hyperbolic periodic point of period q, then tex2html_wrap_inline2788 . If C contains one elliptic periodic point, then tex2html_wrap_inline2792 . If C contains no periodic point, then tex2html_wrap_inline2796 . More generally: if tex2html_wrap_inline1414 is differentiable and for a fixed point tex2html_wrap_inline2760 of tex2html_wrap_inline2726 the matrix tex2html_wrap_inline2804 has no eigenvalue 1, then tex2html_wrap_inline2760 has index 1 if tex2html_wrap_inline2812 and index -1 if tex2html_wrap_inline2816 . (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 tex2html_wrap_inline1380 on tex2html_wrap_inline1436 has n attracting periodic intervals and so n repelling intervals on tex2html_wrap_inline1436 (they all must have index -1), then there exist n sets of index 1 of the same period on each side of tex2html_wrap_inline1436 .

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 tex2html_wrap_inline2838 are related by tex2html_wrap_inline2840 . 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 tex2html_wrap_inline2842 parameterized by points in some interval I. Let C be a homotopically trivial simple closed curve in tex2html_wrap_inline1422 such that for tex2html_wrap_inline2756 , no fixed point of tex2html_wrap_inline2852 is on C and such that C is the boundary of an open subset tex2html_wrap_inline2858 of tex2html_wrap_inline1422 . Since tex2html_wrap_inline2862 is bounded away from 0 and tex2html_wrap_inline1664 Equation 2), there are only finitely many connected fixed point sets in tex2html_wrap_inline2858 . A parameter value tex2html_wrap_inline2756 , for which the number of connected components of fixed points of tex2html_wrap_inline2852 of tex2html_wrap_inline2852 in tex2html_wrap_inline2858 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 tex2html_wrap_inline2756 , there exists at least one fixed point tex2html_wrap_inline2882 of tex2html_wrap_inline2852 in tex2html_wrap_inline2858 . Assume tex2html_wrap_inline2888 is hyperbolic for tex2html_wrap_inline2890 , parabolic for tex2html_wrap_inline2892 and elliptic for tex2html_wrap_inline2894 . If tex2html_wrap_inline2882 is the only fixed point in tex2html_wrap_inline2858 for tex2html_wrap_inline2890 , then tex2html_wrap_inline2902 is a bifurcation value because the index of tex2html_wrap_inline2888 changes from -1 to 1 and the sum of the indices of all fixed points of tex2html_wrap_inline2852 in tex2html_wrap_inline2858 stays constant. If two additional fixed points of index -1 are created at tex2html_wrap_inline2892 , then the bifurcation is a pitchfork bifurcation.
2) Assume, a single fixed point tex2html_wrap_inline2882 exists in tex2html_wrap_inline2858 for tex2html_wrap_inline2890 , and does not exist for tex2html_wrap_inline2894 . 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 tex2html_wrap_inline2892 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 tex2html_wrap_inline1378 varies, the invariant circle tex2html_wrap_inline1384 belonging to the caustic tex2html_wrap_inline1372 moves up in the phase space tex2html_wrap_inline1422 . At each parameter tex2html_wrap_inline1378 , the invariant circle tex2html_wrap_inline1384 divides the phase space tex2html_wrap_inline1422 into a region below tex2html_wrap_inline1436 bounded by tex2html_wrap_inline2960 and tex2html_wrap_inline1384 and a region above tex2html_wrap_inline1384 .

For small tex2html_wrap_inline2966 , the invariant circle tex2html_wrap_inline1384 is near the boundary component tex2html_wrap_inline2960 of the phase space tex2html_wrap_inline1422 . For tex2html_wrap_inline1568 , the invariant circle tex2html_wrap_inline1384 approaches the equator tex2html_wrap_inline2978 of tex2html_wrap_inline1422 . Aubry-Mather sets tex2html_wrap_inline1394 with a fixed rotation number tex2html_wrap_inline2694 pass through the moving circle tex2html_wrap_inline1384 .

The passage of tex2html_wrap_inline1394 with irrational tex2html_wrap_inline2694 is easy to describe: since each parameter value tex2html_wrap_inline2992 with irrational tex2html_wrap_inline2694 is a point of increase of tex2html_wrap_inline1578 , the set of parameters tex2html_wrap_inline1378 for which tex2html_wrap_inline1394 intersects with tex2html_wrap_inline1384 consists of exactly one point, the parameter value tex2html_wrap_inline3004 , for which tex2html_wrap_inline3006 . The set tex2html_wrap_inline1394 is in general a Cantor set.

If the map tex2html_wrap_inline1414 and the curve tex2html_wrap_inline1372 are smooth and tex2html_wrap_inline2694 satisfies a Diophantine condition, then more can be said: both tex2html_wrap_inline1394 and tex2html_wrap_inline1384 depend in a differentiable way on tex2html_wrap_inline1378 (see [4]). If tex2html_wrap_inline1372 is smooth and tex2html_wrap_inline1394 is a smooth invariant curve with Diophantine rotation number tex2html_wrap_inline2694 , then tex2html_wrap_inline1394 is accumulated by other invariant curves filling a set of positive measure [33]. Therefore, if tex2html_wrap_inline3030 is Diophantine, the moving circle tex2html_wrap_inline1384 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 tex2html_wrap_inline1384 if the phase locking interval tex2html_wrap_inline1586 is nontrivial. Since any periodic orbit on tex2html_wrap_inline1384 is a Birkhoff periodic orbit, tex2html_wrap_inline1582 is the set of parameters tex2html_wrap_inline1378 for which M(p/q) intersects tex2html_wrap_inline1384 .



The picture is that for a residual set of caustics tex2html_wrap_inline1372 and a dense set of parameters tex2html_wrap_inline2318 , every Birkhoff periodic orbit on the invariant curve tex2html_wrap_inline1384 is isolated, hyperbolic and has index -1. Since these orbits correspond to periodic orbits of the circle map tex2html_wrap_inline1380 , we can give them coordinates tex2html_wrap_inline3090 and group them in pairs tex2html_wrap_inline3092 . (Compare Figure 11a). Important for us is the following Corollary.



Every q-periodic orbit tex2html_wrap_inline3120 of the billiard map is determined by the values tex2html_wrap_inline3122 and tex2html_wrap_inline3124 is a critical point of the length functional tex2html_wrap_inline3126 , which is the length of the polygon.



Note. We used here again the notation tex2html_wrap_inline3164 , tex2html_wrap_inline3166 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.

tex2html_wrap3496 tex2html_wrap3498



tex2html_wrap3502 tex2html_wrap3504
The same argument proves also:


tex2html_wrap3506 tex2html_wrap3508
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 tex2html_wrap_inline2694 , if there is no invariant circle with rotation number tex2html_wrap_inline2694 . For tex2html_wrap_inline3436 , the non-Birkhoff periodic points can only be above tex2html_wrap_inline1384 and for tex2html_wrap_inline3440 , they can exist only below tex2html_wrap_inline1384 . On the other hand, they can not collide with tex2html_wrap_inline1384 . Thus, the non-Birkhoff periodic orbits appear and disappear in the course of the deformation.
2) We believe that typically, for tex2html_wrap_inline3446 sufficiently near to tex2html_wrap_inline3448 , 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 tex2html_wrap_inline3448 , 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 tex2html_wrap_inline3462 , 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 tex2html_wrap_inline3468 , then all the Birkhoff periodic sets hit the curve tex2html_wrap_inline1384 at the same time tex2html_wrap_inline3472 and tex2html_wrap_inline1384 is exceptional consisting of periodic points.
4) Since by Lemma 2.1, the circle diffeomorphisms tex2html_wrap_inline1380 have derivatives bounded away from zero and tex2html_wrap_inline1664 , there are only finitely many bifurcation points in tex2html_wrap_inline3480 . 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 tex2html_wrap_inline1384 for some tex2html_wrap_inline2318 , there is a set of index 1 leaving tex2html_wrap_inline1384 for some tex2html_wrap_inline3492 . 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.

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