Equidistribution for matings of quadratic maps with the modular group

The paper proves exactly what the candidate claims: for the family Fa on the Klein combination locus K, there exist two canonical probability measures μ+ and μ− supported on the boundaries of the forward/backward limit sets, and for every non-exceptional point z0 the normalized image/preimage counting measures converge weakly to μ+ and μ−, respectively (Theorem 1.2; it also specifies Ea = ∅ unless a=5, when Ea={−1,2}) . The definitions of Fa (via the involution Ja and the deleted covering correspondence CovQ0) and of K match the model’s setup, and the paper records dp(Fa)=dp(Fa−1)=2 . The forward/backward limit sets Λa,± are defined as in the model and are independent of the Klein pair choice up to the smoothing at 1 . The paper also analyzes the exceptional set and shows it is empty unless a=5, where {−1,2} are exceptional, exactly as in the model’s argument (Section 4) . The model additionally states the invariance relations F_*μ+ = 2 μ+ and F^*μ− = 2 μ−; while not written explicitly in Theorem 1.2, these follow immediately by passing to the limit in the weak convergences of (1/2^n)(F^n)_*δz0 and (1/2^n)(F^{-n})_*δz0 (a standard argument in this setting). Overall, the model reproduces the paper’s approach (via Bullett–Lomonaco’s two-sided restriction and conjugacy to a quadratic rational map, then transplanting classical equidistribution), with no substantive divergence.