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.

Figures (7)

• Figure 1: Basic construction of an asymmetric lemon table $Q(b,R)$.
• Figure 2: A bundle of lines generated by $\gamma$, and the cross-section $\sigma$.
• Figure 3: The case with $i_1(x)=0$: there is only one reflection on $\Gamma_R$.
• Figure 4: Subcases of Case 2 according to the values of $\tau_0$. Note that $d_0<pd_0+\frac{d_1}{2}$, which follows from the assumption that $\frac{d_0}{1+i_0}<d_1$.
• Figure 5: First restriction on $R$. The thickened pieces on $\Gamma_1$ are related to $U$.

Theorems & Definitions (27)

• Lemma 1
• Theorem 1
• Theorem 2
• Remark 1
• Conjecture 1
• Definition 1
• Proposition 1
• Proposition 2
• Proposition 3: Do91
• Lemma 2