Counting and equidistribution of strongly reversible closed geodesics in negative curvature

The paper proves that the number of {I,J}-reversible closed geodesics grows like (||σ_I^+||·||σ_J^-||)/(δ_Γ||m_BM||)·exp(δ_Γ T/2), and that for I=J the Lebesgue measures along these orbits equidistribute to m_BM, by reducing to counting/equidistribution of common perpendiculars between fixed-point sets of involutions and carefully handling multiplicities (Theorem 1.1; proof overview and hypotheses) . The key geometric input is that for involutions α,β with disjoint fixed sets, γ=βα is loxodromic with axis generated by the common perpendicular and length λ(γ)=2·d(F_α,F_β) (Lemma 2.3) . The multiset bijections between Γ-orbits of pairs (α,β), common perpendiculars, and reversible conjugacy classes, as well as the multiplicity formula, are explicit (Proposition 3.3 and Eq. (10)) . The counting step is obtained from general theorems on common perpendiculars (Theorem 5.1 specialized to P=0) , and equidistribution follows by decomposing each reversible orbit into two common-perpendicular segments and invoking equidistribution of these segments (Theorem 6.2; identity (29)) . The candidate solution follows the same reduction, uses the same length-halving relation, the same multiplicities, and the same skinning-measure machinery; differences are only presentational. Hence both are correct and essentially the same proof.