SINGULARITY OF COMPOUND STATIONARY MEASURES

The paper states and proves a Main Theorem that, on G = Z2 * Z3 with the word metric, the set of filling measures is not closed under either convolution or convex combination; it exhibits explicit finitely supported pairs µ1, µ2 with both filling while µ1*µ2 and every nontrivial convex combination tµ1+(1−t)µ2 fail to be filling. The negative answers are established through a concrete computation of harmonic measures on the boundary via Denjoy/Minkowski classes, together with explicit algebraic conditions (notably (2.22)–(2.23) and the hyperbola (2.39)) and three worked Theorem-Examples 2.40, 2.41, and 2.42 that directly deliver the desired counterexamples, culminating in the Main Theorem ; see the explicit nearest-neighbor and five-point-support constructions in §2.E–§2.F, including the hyperbola (2.39) and the convolution example using conjugation by a . In contrast, the candidate solution is an incomplete outline: it relies on the h ≤ vℓ inequality and equality characterization to recast filling as h = vℓ, and on analyticity of h and ℓ in order to argue that nontrivial convex combinations or convolutions of filling measures must eventually break equality. However, it does not identify a rigorous pair (or family) for which the “gap function” φ(µ) = h(µ) − vℓ(µ) is proved non-identically zero along the specific segments considered; it appeals to explicit nearest-neighbor formulas on a smaller simplex while the constructed segment lives in a larger finite support; and it claims an explicit µ★ from Mairesse–Mathéus with h = vℓ without supplying the required verification. The paper’s construction is complete and explicit; the model’s argument lacks key details and does not reach a valid proof. The paper’s framework includes: uniqueness of the harmonic measure on ∂G for the walks considered and its concrete parameterization κ_{α,p} (Theorem 2.3) , explicit criteria for membership in the Minkowski class (Theorem 2.21 and, in the nearest-neighbor case, Proposition 2.32 and Corollaries 2.34/2.36) , an identification of the boundary Hausdorff class with a specific κ_{1/2,p} (with p = 1/(1+√2)) , and then the three examples proving non-closure for convex combinations and convolution in §2.F . The conceptual motivation and the relation to maximal entropy/filling in the hyperbolic setting are discussed in §3–§4 (including the role of BHM and Ancona’s theory) , and the strategy for building counterexamples is summarized in §7 .