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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 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 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 on . The polygons 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, .

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.

Remark.
Eq. 3 leads to the question: what kind of a function is , as runs from 1 to ? By eq. 3, for . 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, 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 curve , parametrized by the arc length is completely determined by its radius of curvature . We will assume that 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 .

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

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

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

Let be a convex caustic, and let . Let be the invariant circle, let the induced homeomorphism, and let be a periodic orbit. We assume that the sides of P support the strictly convex arcs of , and be the consecutive supporting points. Let , be the supporting segments (Fig. 10). Denote by , the curvature of T. Let be the arclength elements on respectively. Set . Finally, let be the exterior angles at (see Figure 10).

Remarks.
1) Let be a billiard map, and let be an invariant circle. Let . If , then contains minimal periodic orbits of type (p,q). Periodic points, , satisfy . Typically, they are isolated.
2) If , for a convex caustic , then the one-sided derivatives, , exist, and are given by eq. 5. The jumps of the derivative are due to the flat arcs of . If is strictly convex, then f is , and is given by Eq. 5.
3) With the same assumptions as in the preceeding remark, let be a periodic point, and let P be the corresponding inscribed polygon (Figures 8,10). Let be strictly convex at the supporting points of P. If , then the periodic orbit P consists of hyperbolic points. If , then the points 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 , and an attracting periodic orbit for the circle map if (see Figure 11). If P is parabolic, and isolated, P may be repelling or attracting on each side independently. It is also possible that 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 , 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