Recurrence in the Dynamics of Meromorphic Correspondences and Holomorphic Semigroups

The paper proves a Poincaré recurrence theorem for meromorphic correspondences assuming an F*-invariant probability measure μ that does not charge pluripolar sets, using descriptive set theory to handle the key obstacle that F†(B) need not be Borel; the proof constructs E_n := ⋃_{j≥n}(F^j)†(B) in the completed σ-algebra B_μ, shows μ(F†(E_n)) = μ(E_{n+1}), and concludes μ(B\E)=0 where E are the infinitely recurrent points, yielding a Borel E′⊆E with μ(E′)=μ(B) . The candidate solution gives a different proof: normalize the pullback operator to a Markov operator L=(1/d_t)T, represent it by a Markov kernel Q, build a stationary path-space measure via Ionescu–Tulcea/Kolmogorov, and apply Poincaré recurrence for the shift to get infinitely many returns to B along Γ-chains; translating along the transpose correspondence then yields forward recurrence for μ-a.e. x∈B. The only gap is an implicit identification of G^n(B) with {x: F^n(x)∩B≠∅}; this holds μ-a.e. by the paper’s composition remark (equality off a proper analytic set) and the fact that μ charges no pluripolar sets . Thus both arguments are sound; the paper’s proof is complete and the model’s proof is correct after noting the μ-a.e. equivalence.