Billiards that Share a Triangular Caustic

TL;DRAbstract

We consider a one-parameter family of billiard tables Tℓ which have as a common caustic the equilateral triangle γ. The billiard tables Tℓ are constructed geometrically by the string construction, where the length ℓ of the string is the parameter. We study the family of circle homeomorphisms fℓ obtained by restricting the billiard map to the canonical invariant circle Γℓ belonging to the caustic and the rotation function ρ(ℓ) = ρ(fℓ). We show that the graph of ρ is a devil’s staircase. We analyze the passage of a Birkhoff periodic orbits through the caustic as the parameter changes. 1 A Family of Billiard Maps Let γ be the equilateral triangle with sidelength 1/3. We construct tables T obtained from γ by the so called string construction (see i.e. [11, 12]). One takes an unstretchable string having length ℓ> 1, wraps it around γ, pulls it tight at a point, M, and drags it around γ. The point M then traces the table. Varying the parameter ℓ ∈ [1, ∞], we get a one-parameter family

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