Filled Julia sets of Chebyshev polynomials

The paper proves both statements rigorously: (i) pre-compactness of the filled Julia sets of the dual Chebyshev polynomials together with the inclusion chain K ⊆ Po(K_∞) ⊆ Po(limsup K_n) ⊆ Co(K), and (ii) weak-* convergence of the measures of maximal entropy to the equilibrium measure ω_K. The model’s solution mirrors the high-level goals but contains critical gaps: it incorrectly treats s_n = log|T_n| as 0 on Ω_n \ Ω and appeals to an unproven “stability of Green functions under Hausdorff limits,” using this to claim L^1_loc convergence on all of C and to force K ⊆ Po(K_∞). The paper instead establishes uniform control via a precise inequality (Proposition 3.5) and uses Carathéodory convergence (Proposition 4.3) to obtain the inclusion K ⊆ Po(K_∞); for the measures, it avoids the model’s unproved distributional argument and gives a robust preimage-counting/energy-maximization proof. Hence the paper is correct; the model’s proof is incomplete and relies on unjustified steps.