Periodic and none-periodic spherical billiards

19June1995

PHYSICS

ELSEVIER

LETTERS

A

Physics Letters A 202 (1995) 191-194

Periodic and none-periodic spherical billiards Jian Cheng Department of Mathematics, Nanjing Uniuersity, Nan&g 210008, China Received 24 March 1994; accepted for publication Communicated by A.R. Bishop

19 April 1995

Abstract We introduce a one-parameter family of area preserving maps on an annulus which are derived from billiards on spherical regions. These maps have the property that all orbits are periodic or no orbits are periodic (except one) according to different parameter values. In the latter case each orbit has a dense property.

1. Introduction

2. Notations

It is well known that for area preserving twist maps both hyperbolic and elliptic behavior coexist. For instance, it was shown in Ref. [l] that the Hausdorff dimension of horseshoes in the standard map is close to 2 for large parameter values, and at the same time there are elliptic periodic orbits for some parameter values in a set with infinite measure. In this note we introduce a one-parameter family of area preserving maps on an annulus without twist condition. These maps, which are derived from billiards on some regions of S2, show a very strong elliptic behavior but no hyperbolicity. Also according to different parameter values, the orbits of the map are all periodic or none-periodic (except one). All this seems to suggest that the twist condition is a key factor for the coexistence of hyperbolic and elliptic behavior. For the case of no periodic orbits we show that an orbit of the flow defined by billiards is dense on some region of the billiard table. Two of our main theorems are stated in Section 2 and proved in Sections 3, 4, respectively. In Section 6 we show the ellipticity of the system.

Let S C R3 be a smooth surface with a piecewise smooth boundary 8. Let M be the restriction of the unit tangent bundle on S. A point x = (q, u) E M is called a line element, where q E S is called the supporter of x and u E T4S is a unit tangent vector. Let rr be the natural projection, i.e., rr(x) = q. Note that M is a manifold with a boundary which consists of those x E M with T(X) E XS. By billiards in S we mean the dynamical system of geodesic flow G’ inside S with uniform speed. When a geodesic reaches the boundary 8 an instantaneous reflection from M occurs according to the rule “the angle of incidence equals the angle of reflection”. The system G’ preserves the measure I_Lwith dp = dq d w, where dq is the Euclidean metric on S and do the natural measure on S’. For G’ we construct the Poincare section map. We choose an orientation for as, and for u E T4S an inward unit tangent vector with q E as. These correspond to the arc length s and to an angle 8 from the positive tangent direction on aS to u. s and 0 determine a geodesic inside S. When this geodesic

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and main results

192

J. Cheng / Physics Letters A 202 (199.5) 191-l 94

is an inward unit tangent vector, consider its projection on s’. Let @={G’(x)

--<<
Note that d depends on the line element x, but different line elements will give the same set 6’. 7rb is a subset of S’, cl(r@‘) its closure on S’.

P Fig. 1.

reaches the boundary 8 and after the reflection, one obtains s’, 8’. These values determine this map. An invariant measure of this map is da = c sin 8 ds d 8, where the constant c is the normalization factor. The surface which we consider here is part of the unit sphere. Suppose S is a half-sphere, then 8 is a large circle with length 27r. From the property of the sphere it is easy to see that for any (s, 0) the image under the section map is (s + 7r, 0). Therefore the formula of the map is T: (s, 8)+(s+7r(mod27r),

EM’:

G*(x) and

Theorem 2.2. If 4 = c~rr, where 0 < (Y< 1 is an irrational number, then the system G’ has only one periodic orbit. If x = (q, u) determines an orbit different from the periodic one, then @={G*(x)

EM’:

has the property

--<<
is a region of S’ and

1cl(7r8) I > 0. In order to prove these results we need a special construction, which will be discussed in the next section.

0).

This is an integrable system on an annulus and every point is periodic with period 2, but the twist condition does not exist. Consider a special change on this surface: Let P be a point of ES, and Q E 8 the antipodal point of P. Any geodesic inside S connecting P, Q divides S into two parts (see Fig. 1). We choose one of these two parts as our billiard table, and on this part we assume that 4 is the angle between two boundaries. We denote this part by s’ and choose an orientation for as’. The section map on this part is denoted by T’. Note that this system is only piecewise differentiable since the section map T’ is not defined at P, Q, but geodesics starting from any points different from P, Q will not “die” at P, Q. Our first result about T’ can be stated as follows:

3. Reflection of S’ on the sphere Let us consider s’ as a part of the unit sphere S2. We denote two pieces of the boundary 8’ by yr and y2 respectively (see Fig. 2). Definition 3.1. S,(S’, y2) is said to be the reflection of S’ of yz if S,(S’, y2) is the region on S2 which is bounded by two geodesics y; and y; connecting P, Q such that (1) the angle between y; and yi is 4, (2) yb and y2 coincide but have a different orientation, (3) yr and yi are on different sides of y2 (see Fig. 2). With this concept the reflection of the orbits at the boundary is exactly the same geodesic passing through that piece of the boundary and entering into

Theorem 2.1. If 4 = (m/n)r, with m < n relative prime natural numbers, then all orbits are periodic. The periods of the orbits are the same (except one) and are determined by n: (1) if n is even, then the period is n, (2) if n is odd, then the period is 4n. Let us denote the area of S’ by IS’ I and the restriction of the unit tangent bundle on S’ by M’. If x = (q, u) E ?lM’, where q is neither P nor Q and u

Fig. 2.

J. Cheng / Physics Letters A 202 (1995) 191 -I 94

the region which is the reflection of S’ of the same piece of &S’. This is due to the fact that boundaries are pieces of geodesics which have geodesic curvature zero.

4. The proof of Theorem 2.1 For convenience Theorem

let us restate the theorem.

2.1. If 4 = (m/n>rr,

with m < n relative prime natural numbers, then all orbits are periodic. The periods of the orbits are the same (except one) and are determined by n: (1) if n is even, then the period is n, (2) if n is odd, then the period is 4n. Proof. Clearly, starting from the mid-point of y1 the geodesic which is orthogonal to y1 gives a periodic orbit of period 2 for the section map. Let y be the large circle and

r*=b-rJu{P,Qb Without loss of generality, we only consider the geodesics emerging from y1 and define the following sequence of reflections starting from s’: Let S, be S’ itself. S, is the reflection of S, from y2 and has boundaries y:, y:, where yz and y2 coincide but have a different orientation. S, is the reflection of S, of y: and has boundaries yf, y;, where yf and y: coincide with a different orientation. In general Szr+ 1 is the reflection of S,, and has boundaries yF’+‘, yi’+l, where yF’+’ and y:’ coincide with a different orientation, S,, is the reflection of S,,_ 1 and has boundaries y:‘, yt’, where yil and yz’-l coincide and have a different orientation. (1) It is not hard to see that for 2k y:” is obtained by rotating y1 around the line segment PQ by 2k+ in clockwise direction. Therefore on S, yf and y * coincide the first time and have a different orientation. This indicates that for the section map the orbits have period n. (2) For odd n it is clear that for any k < 2n y: cannot coincide with y1 or y *. y:” coincides with y1 but has an opposite orientation. Therefore one can see that yf” coincides with y1 with the same orientation. This implies that orbits in this case have a period 4n. n

193

5. The proof of Theorem 2.2 and a corollary Let us also state the theorem here first. Theorem 2.2. If do= (an, where 0 < (Y< 1 is an irrational number, then the section map has only one periodic orbit with period 2. Moreover if x = (q, v) determines an orbit different from the periodic one, then

H={G’(x)

EM’:

has the property I cl(?r@) I > 0.

--co<<<} that cl(n8)

is a region on S’ and

Proof Clearly the only periodic orbit is the one orthogonal to both the boundaries y1 and yZ. Let G’(x) be an orbit different from the periodic one. Since (Y is irrational G’(x) can not be periodic. Suppose that x = (q, v> E as’, v E T4S’ is an inward unit tangent vector. This actually determines a large circle on S2, which we denote by g. Let

J=

;it”(q,

P),

where d( , ) is the distance function on S2. We claim that cl(rTTB) = {qES’:

d(q,

P) >Jaand

d(q,

Q) ad}.

It is easy to see that the set on the right hand side of the above equation is a closed region on S’, which we denote by S’(a). By reflection we can see that ~8 c S’(J). Therefore we have cl(n6’) c S’(J). Let qO E S’(ds and &, the smaller values of d(q,, P) and d(q,, Q). Since (Y is irrational, by reflection of S’ of the boundaries infinitely many times, all images of qO on S2 are dense on two closed surves, c, = {q E s2: d(q, c,={q~S~:

P) = &},

d(q,Q)=&}.

Since g crosses these two curves or is tangent to them, any small neighborhood of q,, will have points of ~8. This completes our proof. n

J. Cheng /Physics Letters A 202 (1995) 191 -I 94

194

If G’(x) is an orbit in M’, it corresponds to a sequence ((s,, 0,>):= ~~ for the section map T’ such that (s,,

@J =Tn(s”,

%I).

Let O=

inf 19,, n

then 0 > 0. If {x,} is a sequence of line elements representing different orbits G’(x,) in M’, then for each G’(x,) we have 0,. Corollary

and lim,,,

5.1. If +4= crrr, where (Y is irrational 0, = 0, then

lim Icl(7r~Y~) I = IS’/, n+= where @,,,= (G’( x,) E M’: -x

odic orbits The purpose of studying billiards on S’ was to see if an irrational (Y would produce an ergodic system, not to compare it with degenerate integrable systems. It is well known that billiards on a planar triangle region with an angle satisfying certain conditions might be ergodic, but there is no proof for it. We were looking for “two angle” regions on S2 and trying to establish ergodicity of billiards on such regions. But there is a stable elliptic periodic orbit which prevents the system from being ergodic. Let us complete this article by stating and proving the following: Theorem 6.1. The exceptional period 2 periodic orbit in Theorems 2.1 and 2.2 is stable, therefore it is elliptic.

< t < m}.

Proof. Suppose this is not true, then without loss of generality we may assume that there is an E > 0 such that

IS’1 --E>

6. Closing remarks and the ellipticity of the peri-

lCl(7r@~)I

for all n. This implies that, by using the notation in the proof of Theorem 2.2, there is a J> 0 such that

Proof. Let q, be the mod point of y, and g the large circle passing through q, and perpendicular to yl. Then there is a neighborhood N of (q,, up) in M’, where up is perpendicular to y, at q,, such that every large circle g’ determined by (q, o) E N is very close to g. This gives us the stability of the periodic orbit. Therefore the periodic orbit is elliptic. n

cl( 7r@,t) c S’( 2). Let g, be large circles on S2 representing g the large circle which is tangent to C,={qESY

T@~,, and

Note that a similar argument can be used to show that in the case of C$= (m/n)m all orbits are elliptic.

d(q,P)=d].

Obviously the minimum angle between g and all large circles passing through P, Q is smaller than the minimum angle between g, and all those large circles passing through P, Q, which implies that 0, is not zero. This contradiction completes lim.,, the proof. n

References [l] .I. Cheng, Horseshoes in the standard map, Ph.D. Dissertation, University of Arizona (1992). [2] I.P. Comfeld, S.V. Fomin and Ya.G. Sinai, Ergodic theory (Springer, Berlin, 1981). [3] M. DoCarmo, Differential geometry of curves and surfaces (Prentice Hall, Englewood Cliffs, NJ, 1976).