MATHER β-FUNCTION FOR ELLIPSES AND RIGIDITY

The paper derives β(ρ) for ellipse billiards via a non-standard generating function and invariant measure, obtaining β(ρ) = 2a(√λ/b) − 2√(a^2−λ) E(φ,k) + ρ |E_λ| with ρ = F(φ,k)/(2K(k)) and |E_λ| = 4√(a^2−λ)E(k); see Theorem 2.1, Corollary 2.2, and Theorem 7.1. The candidate solution reproduces the same β-structure but misnormalizes the rotation number as ρ = F(φ,k)/K(k), which doubles it (evident from the limit ρ→1/2 vs 1 as λ→b). This induces an extra factor 2 in the term involving E(k)F(φ,k) in its “equivalent” e,f-form, contradicting the paper’s formula. The paper’s normalization and resulting formulas are internally consistent and supported by the derivations provided.