DYNAMICS OF SEMIGROUPS OF HÉNON-LIKE MAPS IN C2

The paper constructs averaged Green functions G_G^±, proves the semi-invariance identity ∑_{i=1}^{n0} G_G^±∘H_i = D_G G_G^± (hence (1/D_G)∑ H_i^*μ_G^± = μ_G^±), establishes logarithmic growth G_G^+(x,y)=log|y|+c_G^+ on V_R^+ (and analogously for “−”), and identifies Supp(μ_G^±)=J_S^± while showing the semigroup Julia sets equal the closure of the union of individual Julia sets (Theorem 1.3). These steps appear in Corollary 1.2, Corollary 3.2, and the proof of Theorem 1.3 via Harnack-type arguments for harmonic functions, respectively . The candidate’s solution mirrors this strategy: filtration by V_R^± and the estimate D_G ≥ 2n_0 to force uniform Cauchy convergence, the same transfer operator identity, mass normalization via logarithmic growth, and the identification of the Julia set with the Green current’s support, concluding J_S^± = cl(⋃_h J_h^±). The only notable differences are (i) the candidate assumes convergence of G_k^± rather than proving it (the paper gives a detailed proof: Step 1 inequality and uniform convergence on V_R^± and U_S^±, using D_G ≥ 2n_0) , and (ii) a minor notational/torsion point: the paper’s Theorem 1.3 statement says "closure of the union" but displays equality to the union; the proof shows the union equals the semigroup Julia set, hence is closed, resolving the discrepancy . Net: both are correct, with substantially the same proof outline.