Möbius group actions in the solvable chimera model

The paper’s Proposition 9 claims: If μ on S^1 is absolutely continuous, not balanced, and invariant under a non-identity disc automorphism g (μ = g_*μ), then μ must be a Poisson kernel. The proof asserts that g fixes both the conformal barycenter B(μ) and the mean C(μ), hence must be the identity, contradicting nontrivial g; see Proposition 9 and its proof text . However, the argument implicitly uses g(C(μ)) = C(μ), i.e., Möbius-equivariance of the mean, which is false—indeed, the same paper earlier warns that mean values (centroids) are not equivariant under Möbius actions . The model provides a concrete counterexample using an elliptic automorphism of finite order: conjugate a rational rotation R_{2π/q} by a Blaschke map to obtain g; then any rotation-invariant but nonconstant density v(ϕ) (e.g., 1+ε cos(qϕ)) yields an absolutely continuous, non-balanced μ with μ = g_*μ that is not a Poisson kernel. The model also gives the corrected—and true—statement: when g has infinite order (irrational rotation angle), invariance forces the pushed-forward density to be Lebesgue by ergodicity/Fourier arguments, and pulling back produces exactly the Poisson kernel.