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Periodic orbits and their deformations

Birkhoff periodic orbits and invariant circles.
For any rational number there exist at least two distinct Birkhoff periodic orbits of tex2html_wrap_inline1414 with rotation number p/q [18, 15, 16], the periodic orbits of type (p,q) (Figure 9). We denote by |P| the perimeter of any inscribed polygon. For rational p/q, let tex2html_wrap_inline2150 be the space of ordered q-gons, inscribed in T, and winding p times around T. Periodic orbits of type (p,q) are the critical points of the function tex2html_wrap_inline2162 on tex2html_wrap_inline2150 . The polygons tex2html_wrap_inline2166 that maximize |P| are the minimal orbits of type (p,q) because the action is the negative of the length. Besides minimal periodic orbits there are minimax orbits, tex2html_wrap_inline2172 .

  lemma392

trivlist1067

Lemma 4.1 extends obviously to arbitrary area-preserving twist maps, where the action functional replaces the length. Invariant circles with rational rotation number happen for a thin set of twist maps.

   lemma425

  trivlist1070

tex2html_wrap2652 tex2html_wrap2654
Remark.
Eq. 3 leads to the question: what kind of a function is tex2html_wrap_inline2308 , as tex2html_wrap_inline1378 runs from 1 to tex2html_wrap_inline1664 ? By eq. 3, tex2html_wrap_inline2316 for tex2html_wrap_inline2318 . This implies that v is not differentiable near the boundary of a nontrivial phase locking interval because arbitrarily nearby, it has arbitrarily large derivatives. However, tex2html_wrap_inline2322 is monotone and so differentiable almost everywhere. The phenomenon of phase locking is related in general to the lack of differentiability of the minimal action on parameters.

Deformations of minimal periodic orbits.
A strictly convex piecewise tex2html_wrap_inline1406 curve tex2html_wrap_inline1372 , parametrized by the arc length tex2html_wrap_inline2328 is completely determined by its radius of curvature tex2html_wrap_inline2330 . We will assume that tex2html_wrap_inline2332 is piecewise continuous. At each of the finite number of points of discontinuity as well as at the end points of the curve, we assume the existence of the one-sided limits tex2html_wrap_inline2334 .

A closed convex, piecewise tex2html_wrap_inline1406 curve is, by our convention, a finite union of strictly convex tex2html_wrap_inline1406 arcs and straight line segments called flat arcs. Let tex2html_wrap_inline2340 be the subset of piecewise tex2html_wrap_inline1406 caustics.

If tex2html_wrap_inline2344 belongs to a flat arc of tex2html_wrap_inline1372 or if tex2html_wrap_inline2348 belongs to a strictly convex arc and tex2html_wrap_inline2350 on at least one side, we say that A is a flat point of tex2html_wrap_inline1372 . When tex2html_wrap_inline2348 belongs to a strictly convex arc, and tex2html_wrap_inline2358 , we say that A is a vertex or a corner point.

We will freely identify periodic orbits of the billiard map with polygons, tex2html_wrap_inline2362 inscribed in T. A periodic orbit, P is said to admit a deformation, if there exists a continuous deformation tex2html_wrap_inline2368 of periodic orbits P. More precisely if tex2html_wrap_inline2372 are the arclength coordinates, we assume that tex2html_wrap_inline2374 are continuous. We will also consider one-sided deformations P(t), tex2html_wrap_inline2378 , or tex2html_wrap_inline2380 , assuming the same conditions, except the derivatives at t=0 are one-sided.

tex2html_wrap2656 tex2html_wrap2658
Let tex2html_wrap_inline1372 be a convex caustic, and let tex2html_wrap_inline2390 . Let tex2html_wrap_inline2392 be the invariant circle, let tex2html_wrap_inline2394 the induced homeomorphism, and let tex2html_wrap_inline2396 be a periodic orbit. We assume that the sides of P support the strictly convex arcs of tex2html_wrap_inline1372 , and tex2html_wrap_inline2402 be the consecutive supporting points. Let tex2html_wrap_inline2404 , be the supporting segments (Fig. 10). Denote by tex2html_wrap_inline2406 , the curvature of T. Let tex2html_wrap_inline2410 be the arclength elements on tex2html_wrap_inline2412 respectively. Set tex2html_wrap_inline2414 . Finally, let tex2html_wrap_inline2416 be the exterior angles at tex2html_wrap_inline2418 (see Figure 10).

     lemma487

     trivlist1073

tex2html_wrap2660 tex2html_wrap2662

  lemma571

trivlist1076

  lemma581

trivlist1079

  coro594

trivlist1082

Remarks.
1) Let tex2html_wrap_inline1700 be a billiard map, and let tex2html_wrap_inline1428 be an invariant circle. Let tex2html_wrap_inline2394 . If tex2html_wrap_inline2588 , then tex2html_wrap_inline1436 contains minimal periodic orbits of type (p,q). Periodic points, tex2html_wrap_inline2594 , satisfy tex2html_wrap_inline2596 . Typically, they are isolated.
2) If tex2html_wrap_inline2598 , for a convex caustic tex2html_wrap_inline1372 , then the one-sided derivatives, tex2html_wrap_inline2602 , exist, and are given by eq. 5. The jumps of the derivative are due to the flat arcs of tex2html_wrap_inline1372 . If tex2html_wrap_inline1372 is strictly convex, then f is tex2html_wrap_inline1406 , and tex2html_wrap_inline2612 is given by Eq. 5.
3) With the same assumptions as in the preceeding remark, let tex2html_wrap_inline2594 be a periodic point, and let P be the corresponding inscribed polygon (Figures 8,10). Let tex2html_wrap_inline1372 be strictly convex at the supporting points of P. If tex2html_wrap_inline2622 , then the periodic orbit P consists of hyperbolic points. If tex2html_wrap_inline2626 , then the points tex2html_wrap_inline2628 are parabolic. By Lemma 4.3, the condition that P be parabolic is necessary for the existence of periodic deformations of P. A stronger assertion holds: if P is not an isolated periodic orbit, then P is parabolic.

If P is a hyperbolic periodic orbit, then it is a repelling periodic orbit for the circle map if tex2html_wrap_inline2640 , and an attracting periodic orbit for the circle map if tex2html_wrap_inline2642 (see Figure 11). If P is parabolic, and isolated, P may be repelling or attracting on each side independently. It is also possible that tex2html_wrap_inline1436 contains isolated periodic points, and nontrivial intervals of periodic points as well (see Figure 11 b)). There is a similar analysis of the one-sided derivatives of tex2html_wrap_inline2650 , and a natural extension of eq. 5.

Next: Aubry-Mather sets and the Up: Billiards that share a Previous: Devil's staircase

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