A DIRECT PROOF OF THE EXISTENCE OF MME FOR FINITE HORIZON SINAI BILLIARDS

Carrand proves exactly the points the candidate flagged as sticking: (i) weak-* closedness/compactness of M(M,T) via a suspension-flow argument (Proposition 3.1) ; (ii) upper semicontinuity of μ ↦ h_μ(T)+∫g dμ for piecewise-continuous g under negligible singularities (Lemma 2.1 and Proposition 2.2) ; (iii) a direct variational principle for the discontinuous billiard map (Proposition 2.3) ; and (iv) genericity (dense Gδ) of negligible singularities (Lemma 4.5 and Proposition 4.6) . Theorem 1.1 then yields existence of equilibrium states, including MME for the map and flow, under negligible singularities, and compactness of M(M,T) . Hence the candidate’s “likely open as of cutoff” assessment is superseded by the paper’s correct results.