Chaotic lensed billiards

The paper’s Theorem 9 states exactly the identities the candidate proves: ⟨N⟩_E = A/a, ⟨T⟩_E = (n√π Γ(n/2+1/2)/[s Γ(n/2+1)])·V/a, ⟨τ⟩_V = (n√π Γ(n/2+1/2)/[s Γ(n/2+1)])·V/A, the factorization ⟨T⟩_E = ⟨N⟩_E⟨τ⟩_V, and the E0-claim that the mean number of returns to E before hitting E0 is 1/r0^{n−1} . The paper proves (1) and (4) via ergodic averages and (2) by a collar-region calculation, deducing (3) from the others , and uses the cosine-law measure to get the E0 result . The candidate’s proof uses a Kac/inducing tower identity for returns to E and a flux/coarea computation for ⟨τ⟩_V. Aside from a small bookkeeping slip with the factor s in the unnormalized Liouville measure (it cancels in the final ratio), the candidate’s derivation is correct and yields the same formulas by a different route.