Existence of stable periodic orbits in billiards close to lemon and moon billiards

Abstract

It is known that at lemon and moon billiards that have a sufficiently small curvature on one of their circular arcs are hyperbolic. In this paper we show that replacing this circular arc by a more general boundary component of small curvature could produce billiard tables that admit nonlinearly stable periodic orbits.

Figures (8)

• Figure 1: Examples of Sinai billiards (left) and Bunimovich billiards (right).
• Figure 2: Symmetric flower billiard with two petals and the corresponding folded flower table.
• Figure 3: Modifications of the folded flower billiard to a lemon billiard and to a moon billiard.
• Figure 4: Perturbations of the folded two-petal flower billiard.
• Figure 5: Construction of a billiard orbit segment (left figure) that will be used to produce an elliptic periodic orbit of period $N+2$ (right figure).

Theorems & Definitions (8)

• Theorem 1.1
• Lemma 3.1
• Lemma 3.2
• proof
• Proposition 3.3
• Remark 3.4
• Remark 4.1
• Remark 4.2