Geometric and Combinatorial Properties of Self-similar Multifractal Measures

The paper’s Theorem 4.5 establishes precisely the three requested conclusions for ν (complete multifractal formalism; density of periodic-point local dimensions; identification of D(ν); plus τ(ν,q) independence and τ(µp,q)=τ(ν,q) for q≥0), using the net-interval/transition-graph machinery and a transfer from Feng–Lau’s result on maximal balls U0 to essential net intervals . The matrix-cocycle identity Qp(∆0)T(η)=Qp(∆m) gives exact control of masses along admissible paths , the WSC yields a unique essential class and bounded combinatorics , and µp(K\Kess)=0 supports the τ(µp,q)=τ(ν,q) step for q≥0 . The candidate solution presents a different, pressure-based sketch on the essential class (subadditive pressure P(q,s), Carathéodory construction, and periodic-orbit approximation), which is conceptually consistent with the paper and reaches the same conclusions. Some technical ingredients in the model’s sketch (existence of the pressure limit and equilibrium/Gibbs measures on a countable irreducible graph) are asserted but not derived, whereas the paper avoids these by reduction to U0 and matrix transfer. Overall, the claims match and the approaches differ, so the proper verdict is that both are correct with different proofs.