Caustic-Free Regions for Billiards in the Hyperbolic Plane

The paper proves two results: (1) any convex caustic lies in a uniform collar whose explicit thickness ε depends on D and κ_min (Theorem 1.1), and (2) a jump in curvature creates a caustic-free collar near the boundary (Theorem 1.2). The core ingredients are the hyperbolic mirror equation 1/tanh(a)+1/tanh(b)=2κ/ sin(θ) and the string (Lazutkin) construction. The proof of (1) proceeds via two precise lemmas: tanh^2(δ_γ)/tanh(D) < L (Lemma 3.3) and an explicit upper bound on the Lazutkin parameter L in terms of D and κ_min (Lemma 3.4), which together yield Proposition 3.1 and hence Theorem 1.1 with the stated ε . For (2), the paper constructs a forbidden rectangle in phase space near a curvature jump and then uses Birkhoff’s theorem and a no-crossing argument to force a caustic-free collar (Theorem 1.2) . By contrast, the candidate solution misstates the key quantitative steps for (1): it asserts L ≤ C κ_min^{-1/2}… and tanh ρ ≤ C L, whereas the paper proves tanh^2(δ_γ)/tanh D < L and L < 2κ(m)^2 tanh^2(D) sinh(D). These errors lead to incorrect dependence on κ_min and the wrong functional relationship between δ_γ and L. The qualitative strategy for (2) matches the paper, but (1) is quantitatively incorrect.