On Another Edge of Defocusing: Hyperbolicity of Asymmetric Lemon Billiards

Abstract

Defocusing mechanism provides a way to construct chaotic (hyperbolic) billiards with focusing components by separating all regular components of the boundary of a billiard table sufficiently far away from each focusing component. If all focusing components of the boundary of the billiard table are circular arcs, then the above separation requirement reduces to that all circles obtained by completion of focusing components are contained in the billiard table. In the present paper we demonstrate that a class of convex tables-asymmetric lemons, whose boundary consists of two circular arcs, generate hyperbolic billiards. This result is quite surprising because the focusing components of the asymmetric lemon table are extremely close to each other, and because these tables are perturbations of the first convex ergodic billiard constructed more than forty years ago.2000 Mathematics Subject Classification. 37D50.

Ergodicity in Umbrella Billiards

“…In [25] symmetric lemon billiards were considered, and recently a class of asymmetric lemon-shaped convex billiard tables were constructed in [13], obtained by intersection of two disks in the plane. It was also proved that a subclass of these billiards are indeed hyperbolic using continued fraction techniques [12]. These, along with the moon billiards recently investigated by the authors in [16], are the direct antecedents of the current investigation.…”

“…Conversely, when B approaches 0 and the table approaches the circular case, a host of elliptic islands emerge, becoming long and narrow approaching the integrable case. If R = 1, however, and asymmetric billiards are considered, hyperbolicity [12] and apparent ergodicity [13] often arise, as in Region I in Figure 7.…”

Some new surprises in chaos

“…However, the corresponding billiard is chaotic (hyperbolic). 22 Moreover, this class of billiards has another striking feature. A basic fascinating property of chaotic (hyperbolic) dynamical systems is that local exponential instability (hyperbolicity) ensures global chaotic properties of dynamics like positivity of Kolmogorov-Sinai entropy, mixing (decay of correlations) on each ergodic component, etc.…”

“…However, the fact that only this clearly local condition precludes the existence of other KAM islands in the phase space still looks like a miracle, although proven. 22 Observe that the billiard table in Fig. 12(c) can be considered an asymmetric lemon (Fig.…”

Convex billiards on convex spheres

“…Since then, the mathematical study of chaotic billiards has developed at a remarkable speed, and the defocusing mechanism for chaos were discovered by Bunimovich [Bun74,Bun92], Wojtkowski [Woj86], Markarian [Mar88] and Donnay [Don91]. Very recently, the dynamics of some asymmetric lemon billiards are proved to be hyperbolic [BZZ14], for which the separation condition in the defocusing mechanism was strongly violated. See [Vet84,KSS89,GSG99] for the study of chaotic billiards on general surfaces.…”